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Preliminaries

Lorentz Form (metric) applied to a vector (n\times 1 matrix) produces a co-vector (1\times n matrix). Scalar and tensor products use matrix multiplication.

fricas
ID:=diagonalMatrix [1,1,1,1];
Type: Matrix(Integer)
fricas
G:=diagonalMatrix [-1,1,1,1]

\label{eq1}\left[ 
\begin{array}{cccc}
- 1 & 0 & 0 & 0 
\
0 & 1 & 0 & 0 
\
0 & 0 & 1 & 0 
\
0 & 0 & 0 & 1 
(1)
Type: Matrix(Integer)
fricas
Scalar := Expression Integer

\label{eq2}\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ })(2)
Type: Type
fricas
vect(x:List Scalar):Matrix Scalar == matrix map(y+->[y],x)
Function declaration vect : List(Expression(Integer)) -> Matrix( Expression(Integer)) has been added to workspace.
Type: Void
fricas
g(x:Matrix Scalar):Matrix Scalar == transpose(x)*G
Function declaration g : Matrix(Expression(Integer)) -> Matrix( Expression(Integer)) has been added to workspace.
Type: Void
fricas
dot(x:Matrix Scalar,y:Matrix Scalar):Scalar == g(x)*y 
Function declaration dot : (Matrix(Expression(Integer)),Matrix( Expression(Integer))) -> Expression(Integer) has been added to workspace.
Type: Void
fricas
tensor(x:Matrix Scalar,y:Matrix Scalar):Matrix Scalar == x*g(y)
Function declaration tensor : (Matrix(Expression(Integer)),Matrix( Expression(Integer))) -> Matrix(Expression(Integer)) has been added to workspace.
Type: Void

Verification

fricas
htrigs2exp == rule
  cosh(a) == (exp(a)+exp(-a))/2
  sinh(a) == (exp(a)-exp(-a))/2
Type: Void
fricas
sinhcosh == rule
  ?c*exp(a)+?c*exp(-a) == 2*c*cosh(a)
  ?c*exp(a)-?c*exp(-a) == 2*c*sinh(a)
  ?c*exp(a-b)+?c*exp(b-a) == 2*c*cosh(a-b)
  ?c*exp(a-b)-?c*exp(b-a) == 2*c*sinh(a-b)
Type: Void
fricas
expandhtrigs == rule
  cosh(:x+y) == sinh(x)*sinh(y)+cosh(x)*cosh(y)
  sinh(:x+y) == cosh(x)*sinh(y)+sinh(x)*cosh(y)
  cosh(2*x) == 2*cosh(x)^2-1
  sinh(2*x) == 2*sinh(x)*cosh(x)
Type: Void
fricas
expandhtrigs2 == rule
  cosh(2*x+2*y) == 2*cosh(x+y)^2-1
  sinh(2*x+2*y) == 2*sinh(x+y)*cosh(x+y)
  cosh(2*x-2*y) == 2*cosh(x-y)^2-1
  sinh(2*x-2*y) == 2*sinh(x-y)*cosh(x-y)
Type: Void
fricas
Simplify(x:Scalar):Scalar == htrigs sinhcosh simplify htrigs2exp x
Function declaration Simplify : Expression(Integer) -> Expression( Integer) has been added to workspace.
Type: Void
fricas
possible(x)==subst(x, map(y+->(y=(random(100) - random(100))),variables x) )
Type: Void
fricas
is?(eq:Equation Scalar):Boolean == (Simplify(lhs(eq)-rhs(eq))=0)::Boolean
Function declaration is? : Equation(Expression(Integer)) -> Boolean has been added to workspace.
Type: Void
fricas
Is?(eq:Equation(Matrix(Scalar))):Boolean == _
(map(Simplify,lhs(eq)-rhs(eq)) :: Matrix Expression AlgebraicNumber = _
zero(nrows(lhs(eq)),ncols(lhs(eq)))$Matrix Expression AlgebraicNumber )::Boolean
Function declaration Is? : Equation(Matrix(Expression(Integer))) -> Boolean has been added to workspace.
Type: Void

Massive Objects

A material object (also referred to as an observer) is represented by a time-like 4-vector

fricas
P:=vect [p0,p1,p2,p3]
fricas
Compiling function vect with type List(Expression(Integer)) -> 
      Matrix(Expression(Integer))

\label{eq3}\left[ 
\begin{array}{c}
p 0 
\
p 1 
\
p 2 
\
p 3 
(3)
Type: Matrix(Expression(Integer))
fricas
dot(P,P)
fricas
Compiling function g with type Matrix(Expression(Integer)) -> Matrix
      (Expression(Integer))
fricas
Compiling function dot with type (Matrix(Expression(Integer)),Matrix
      (Expression(Integer))) -> Expression(Integer)

\label{eq4}{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}(4)
Type: Expression(Integer)
fricas
solve(%=-1,p0)

\label{eq5}\left[{p 0 ={\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}}, \:{p 0 = -{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}}\right](5)
Type: List(Equation(Expression(Integer)))
fricas
Q:=vect [q0,q1,q2,q3];
Type: Matrix(Expression(Integer))
fricas
S:=1/sqrt(1-s1^2-s2^2-s3^2)*vect [1,-s1,-s2,-s3]

\label{eq6}\left[ 
\begin{array}{c}
{1 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
-{s 1 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
-{s 2 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
-{s 3 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
(6)
Type: Matrix(Expression(Integer))
fricas
dot(S,S)

\label{eq7}- 1(7)
Type: Expression(Integer)
fricas
T:=1/sqrt(1-t1^2-t2^2-t3^2)*vect [1,-t1,-t2,-t3];
Type: Matrix(Expression(Integer))
fricas
W:=1/sqrt(1-w1^2-w2^2-w3^2)*vect [1,-w1,-w2,-w3];
Type: Matrix(Expression(Integer))
fricas
U:=vect [cosh(u),sinh(u),0,0]

\label{eq8}\left[ 
\begin{array}{c}
{\cosh \left({u}\right)}
\
{\sinh \left({u}\right)}
\
0 
\
0 
(8)
Type: Matrix(Expression(Integer))
fricas
simplify dot(U,U)

\label{eq9}- 1(9)
Type: Expression(Integer)
fricas
V:=vect [cosh(v),sinh(v),0,0];
Type: Matrix(Expression(Integer))

Massless Photons

A photon is a represented by a light-like null 4-vector

fricas
A:=vect [a0,a0*a1,a0*a2,a0*a3]

\label{eq10}\left[ 
\begin{array}{c}
a 0 
\
{a 0 \  a 1}
\
{a 0 \  a 2}
\
{a 0 \  a 3}
(10)
Type: Matrix(Expression(Integer))
fricas
solve(dot(A,A)=0,a3)

\label{eq11}\left[{a 3 ={\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}, \:{a 3 = -{\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}\right](11)
Type: List(Equation(Expression(Integer)))
fricas
A:=vect [a0,a0*a1,a0*a2,a0*sqrt(1-a1^2-a2^2)]

\label{eq12}\left[ 
\begin{array}{c}
a 0 
\
{a 0 \  a 1}
\
{a 0 \  a 2}
\
{a 0 \ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}
(12)
Type: Matrix(Expression(Integer))
fricas
dot(A,A)

\label{eq13}0(13)
Type: Expression(Integer)
fricas
B:=vect [b0,b0*b1,b0*b2,b0*sqrt(1-b1^2-b2^2)]

\label{eq14}\left[ 
\begin{array}{c}
b 0 
\
{b 0 \  b 1}
\
{b 0 \  b 2}
\
{b 0 \ {\sqrt{-{{b 2}^{2}}-{{b 1}^{2}}+ 1}}}
(14)
Type: Matrix(Expression(Integer))
fricas
C:=vect [c0,c0*c1,c0*c2,c0*sqrt(1-c1^2-c2^2)]

\label{eq15}\left[ 
\begin{array}{c}
c 0 
\
{c 0 \  c 1}
\
{c 0 \  c 2}
\
{c 0 \ {\sqrt{-{{c 2}^{2}}-{{c 1}^{2}}+ 1}}}
(15)
Type: Matrix(Expression(Integer))

Observer "at rest"

fricas
R:=vect [1,0,0,0]

\label{eq16}\left[ 
\begin{array}{c}
1 
\
0 
\
0 
\
0 
(16)
Type: Matrix(Expression(Integer))
fricas
dot(R,R)

\label{eq17}- 1(17)
Type: Expression(Integer)

Associated with each such vector is the orthogonal 3-d Euclidean subspace E_P =\{x | P \cdot x = 0\}

Relative Velocity

An object P has a unique relative velocity ω(P,Q) with respect to object Q given by

fricas
ω(P,Q)==-P/dot(P,Q)-Q
Type: Void
fricas
map(Simplify, ω(P,Q))
fricas
Compiling function ω with type (Matrix(Expression(Integer)),Matrix(
      Expression(Integer))) -> Matrix(Expression(Integer))
fricas
Compiling body of rule htrigs2exp to compute value of type Ruleset(
      Integer,Integer,Expression(Integer))
fricas
Compiling body of rule sinhcosh to compute value of type Ruleset(
      Integer,Integer,Expression(Integer))
fricas
Compiling function Simplify with type Expression(Integer) -> 
      Expression(Integer)

\label{eq18}\left[ 
\begin{array}{c}
{{-{p 3 \  q 0 \  q 3}-{p 2 \  q 0 \  q 2}-{p 1 \  q 0 \  q 1}+{p 0 \ {{q 0}^{2}}}- p 0}\over{{p 3 \  q 3}+{p 2 \  q 2}+{p 1 \  q 1}-{p 0 \  q 0}}}
\
{{-{p 3 \  q 1 \  q 3}-{p 2 \  q 1 \  q 2}-{p 1 \ {{q 1}^{2}}}+{p 0 \  q 0 \  q 1}- p 1}\over{{p 3 \  q 3}+{p 2 \  q 2}+{p 1 \  q 1}-{p 0 \  q 0}}}
\
{{-{p 3 \  q 2 \  q 3}-{p 2 \ {{q 2}^{2}}}+{{\left(-{p 1 \  q 1}+{p 0 \  q 0}\right)}\  q 2}- p 2}\over{{p 3 \  q 3}+{p 2 \  q 2}+{p 1 \  q 1}-{p 0 \  q 0}}}
\
{{-{p 3 \ {{q 3}^{2}}}+{{\left(-{p 2 \  q 2}-{p 1 \  q 1}+{p 0 \  q 0}\right)}\  q 3}- p 3}\over{{p 3 \  q 3}+{p 2 \  q 2}+{p 1 \  q 1}-{p 0 \  q 0}}}
(18)
Type: Matrix(Expression(Integer))
fricas
map(Simplify, ω(P,R))

\label{eq19}\left[ 
\begin{array}{c}
0 
\
{p 1 \over p 0}
\
{p 2 \over p 0}
\
{p 3 \over p 0}
(19)
Type: Matrix(Expression(Integer))
fricas
map(Simplify, ω(S,T))

\label{eq20}\left[ 
\begin{array}{c}
{{{{t 3}^{2}}-{s 3 \  t 3}+{{t 2}^{2}}-{s 2 \  t 2}+{{t 1}^{2}}-{s 1 \  t 1}}\over{{\left({s 3 \  t 3}+{s 2 \  t 2}+{s 1 \  t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}}
\
{{-{s 1 \ {{t 3}^{2}}}+{s 3 \  t 1 \  t 3}-{s 1 \ {{t 2}^{2}}}+{s 2 \  t 1 \  t 2}- t 1 + s 1}\over{{\left({s 3 \  t 3}+{s 2 \  t 2}+{s 1 \  t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}}
\
{{-{s 2 \ {{t 3}^{2}}}+{s 3 \  t 2 \  t 3}+{{\left({s 1 \  t 1}- 1 \right)}\  t 2}-{s 2 \ {{t 1}^{2}}}+ s 2}\over{{\left({s 3 \  t 3}+{s 2 \  t 2}+{s 1 \  t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}}
\
{{{{\left({s 2 \  t 2}+{s 1 \  t 1}- 1 \right)}\  t 3}-{s 3 \ {{t 2}^{2}}}-{s 3 \ {{t 1}^{2}}}+ s 3}\over{{\left({s 3 \  t 3}+{s 2 \  t 2}+{s 1 \  t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}}
(20)
Type: Matrix(Expression(Integer))
fricas
map(Simplify, ω(S,R))

\label{eq21}\left[ 
\begin{array}{c}
0 
\
- s 1 
\
- s 2 
\
- s 3 
(21)
Type: Matrix(Expression(Integer))
fricas
map(Simplify, ω(U,V))

\label{eq22}\left[ 
\begin{array}{c}
{{-{\cosh \left({{2 \  v}- u}\right)}+{\cosh \left({u}\right)}}\over{2 \ {\cosh \left({v - u}\right)}}}
\
{{-{\sinh \left({{2 \  v}- u}\right)}+{\sinh \left({u}\right)}}\over{2 \ {\cosh \left({v - u}\right)}}}
\
0 
\
0 
(22)
Type: Matrix(Expression(Integer))

Idempotent Observers

fricas
PP:=tensor(-P,P)
fricas
Compiling function tensor with type (Matrix(Expression(Integer)),
      Matrix(Expression(Integer))) -> Matrix(Expression(Integer))

\label{eq23}\left[ 
\begin{array}{cccc}
{{p 0}^{2}}& -{p 0 \  p 1}& -{p 0 \  p 2}& -{p 0 \  p 3}
\
{p 0 \  p 1}& -{{p 1}^{2}}& -{p 1 \  p 2}& -{p 1 \  p 3}
\
{p 0 \  p 2}& -{p 1 \  p 2}& -{{p 2}^{2}}& -{p 2 \  p 3}
\
{p 0 \  p 3}& -{p 1 \  p 3}& -{p 2 \  p 3}& -{{p 3}^{2}}
(23)
Type: Matrix(Expression(Integer))
fricas
QQ:=tensor(-Q,Q);
Type: Matrix(Expression(Integer))
fricas
is?(trace(PP*QQ)=dot(P,Q)^2)
fricas
Compiling function is? with type Equation(Expression(Integer)) -> 
      Boolean

\label{eq24} \mbox{\rm true} (24)
Type: Boolean
fricas
RR:=tensor(-R,R)

\label{eq25}\left[ 
\begin{array}{cccc}
1 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
(25)
Type: Matrix(Expression(Integer))
fricas
SS:=map(Simplify,tensor(-S,S))

\label{eq26}\left[ 
\begin{array}{cccc}
-{1 \over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}& -{s 1 \over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}& -{s 2 \over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}& -{s 3 \over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}
\
{s 1 \over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{{s 1}^{2}}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{s 1 \  s 2}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{s 1 \  s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}
\
{s 2 \over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{s 1 \  s 2}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{{s 2}^{2}}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{s 2 \  s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}
\
{s 3 \over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{s 1 \  s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{s 2 \  s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}&{{{s 3}^{2}}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}- 1}}
(26)
Type: Matrix(Expression(Integer))
fricas
Is?(SS*SS=SS)
fricas
Compiling function Is? with type Equation(Matrix(Expression(Integer)
      )) -> Boolean

\label{eq27} \mbox{\rm true} (27)
Type: Boolean
fricas
trace(SS)

\label{eq28}1(28)
Type: Expression(Integer)
fricas
TT:=map(Simplify,tensor(-T,T));
Type: Matrix(Expression(Integer))
fricas
Is?(SS*TT*SS = dot(S,T)^2 * SS)

\label{eq29} \mbox{\rm true} (29)
Type: Boolean
fricas
UU:=map(Simplify,tensor(-U,U))

\label{eq30}\left[ 
\begin{array}{cccc}
{{{\cosh \left({2 \  u}\right)}+ 1}\over 2}& -{{\sinh \left({2 \  u}\right)}\over 2}& 0 & 0 
\
{{\sinh \left({2 \  u}\right)}\over 2}&{{-{\cosh \left({2 \  u}\right)}+ 1}\over 2}& 0 & 0 
\
0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
(30)
Type: Matrix(Expression(Integer))
fricas
VV:=map(Simplify,tensor(-V,V));
Type: Matrix(Expression(Integer))
fricas
map(Simplify, UU*VV)

\label{eq31}\left[ 
\begin{array}{cccc}
{{{\cosh \left({2 \  v}\right)}+{\cosh \left({{2 \  v}-{2 \  u}}\right)}+{\cosh \left({2 \  u}\right)}+ 1}\over 4}&{{-{\sinh \left({2 \  v}\right)}-{\sinh \left({{2 \  v}-{2 \  u}}\right)}-{\sinh \left({2 \  u}\right)}}\over 4}& 0 & 0 
\
{{{\sinh \left({2 \  v}\right)}-{\sinh \left({{2 \  v}-{2 \  u}}\right)}+{\sinh \left({2 \  u}\right)}}\over 4}&{{-{\cosh \left({2 \  v}\right)}+{\cosh \left({{2 \  v}-{2 \  u}}\right)}-{\cosh \left({2 \  u}\right)}+ 1}\over 4}& 0 & 0 
\
0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
(31)
Type: Matrix(Expression(Integer))
fricas
WW:=map(Simplify,tensor(-W,W));
Type: Matrix(Expression(Integer))
fricas
Is?(SS*TT*WW = -dot(S,T)*dot(T,W)/dot(S,W)*SS*WW)

\label{eq32} \mbox{\rm true} (32)
Type: Boolean
fricas
Is?(SS*TT*SS = dot(S,T)^2*SS)

\label{eq33} \mbox{\rm true} (33)
Type: Boolean

Nilpotent Operators

fricas
AA:=tensor(-A,A)

\label{eq34}\left[ 
\begin{array}{cccc}
{{a 0}^{2}}& -{{{a 0}^{2}}\  a 1}& -{{{a 0}^{2}}\  a 2}& -{{{a 0}^{2}}\ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}
\
{{{a 0}^{2}}\  a 1}& -{{{a 0}^{2}}\ {{a 1}^{2}}}& -{{{a 0}^{2}}\  a 1 \  a 2}& -{{{a 0}^{2}}\  a 1 \ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}
\
{{{a 0}^{2}}\  a 2}& -{{{a 0}^{2}}\  a 1 \  a 2}& -{{{a 0}^{2}}\ {{a 2}^{2}}}& -{{{a 0}^{2}}\  a 2 \ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}
\
{{{a 0}^{2}}\ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}& -{{{a 0}^{2}}\  a 1 \ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}& -{{{a 0}^{2}}\  a 2 \ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}&{{{{a 0}^{2}}\ {{a 2}^{2}}}+{{{a 0}^{2}}\ {{a 1}^{2}}}-{{a 0}^{2}}}
(34)
Type: Matrix(Expression(Integer))
fricas
BB:=tensor(-B,B);
Type: Matrix(Expression(Integer))
fricas
CC:=tensor(-C,C);
Type: Matrix(Expression(Integer))
fricas
Is?(AA*AA=0*AA)

\label{eq35} \mbox{\rm true} (35)
Type: Boolean
fricas
trace(AA)

\label{eq36}0(36)
Type: Expression(Integer)
fricas
is?(trace(AA*BB)=dot(A,B)^2)

\label{eq37} \mbox{\rm true} (37)
Type: Boolean
fricas
dot(A,B)

\label{eq38}\begin{array}{@{}l}
\displaystyle
{a 0 \  b 0 \ {\sqrt{-{{b 2}^{2}}-{{b 1}^{2}}+ 1}}\ {\sqrt{-{{a 2}^{2}}-{{a 1}^{2}}+ 1}}}+{a 0 \  a 2 \  b 0 \  b 2}+ 
\
\
\displaystyle
{a 0 \  a 1 \  b 0 \  b 1}-{a 0 \  b 0}
(38)
Type: Expression(Integer)
fricas
possible(%)::Complex Float
fricas
Compiling function possible with type Expression(Integer) -> 
      Expression(Integer)

\label{eq39}-{718.6358371582 \<u> 96116}(39)
Type: Complex(Float)
fricas
Is?(AA*BB*CC = -dot(A,B)*dot(B,C)/dot(A,C)*AA*CC)

\label{eq40} \mbox{\rm true} (40)
Type: Boolean
fricas
Is?(AA*BB*SS = -dot(A,B)*dot(B,S)/dot(A,S)*AA*SS)

\label{eq41} \mbox{\rm true} (41)
Type: Boolean
fricas
Is?(AA*BB*SS=SS*AA*BB)

\label{eq42} \mbox{\rm false} (42)
Type: Boolean
fricas
Is?(AA*BB*AA = dot(A,B)^2*AA)

\label{eq43} \mbox{\rm true} (43)
Type: Boolean

Lie Bracket

fricas
STW:=(SS*TT-TT*SS)*WW;
Type: Matrix(Expression(Integer))
fricas
Is?(STW*STW=0*STW)

\label{eq44} \mbox{\rm true} (44)
Type: Boolean
fricas
trace(STW)

\label{eq45}0(45)
Type: Expression(Integer)
fricas
solve(map(x+->x=0,members(STW-AA)),[a0,a1,a2,a3])

\label{eq46}\left[{\left[ \right]}\right](46)
Type: List(List(Equation(Expression(Integer))))
fricas
ABC:=(AA*BB-BB*AA)*CC;
Type: Matrix(Expression(Integer))
fricas
Is?(ABC*ABC=0*ABC)

\label{eq47} \mbox{\rm true} (47)
Type: Boolean
fricas
trace(ABC)

\label{eq48}0(48)
Type: Expression(Integer)
fricas
PQR:=(PP*QQ-PP*QQ)*tensor(-vect([r0,r1,r2,r3]),vect([r0,r1,r2,r3]));
Type: Matrix(Expression(Integer))
fricas
Is?(PQR*PQR=0*PQR)

\label{eq49} \mbox{\rm true} (49)
Type: Boolean
fricas
trace(PQR)

\label{eq50}0(50)
fricas
map(Simplify,(PP*QQ-QQ*PP)*(PP*QQ-QQ*PP))

\label{eq51}\left[ 
\begin{array}{cccc}
{{{{p 0}^{2}}\ {{p 3}^{2}}\ {{q 3}^{4}}}+{{\left({2 \ {{p 0}^{2}}\  p 2 \  p 3 \  q 2}+{2 \ {{p 0}^{2}}\  p 1 \  p 3 \  q 1}+{{\left(-{2 \  p 0 \ {{p 3}^{3}}}-{2 \ {{p 0}^{3}}\  p 3}\right)}\  q 0}\right)}\ {{q 3}^{3}}}+{{\left({{\left({{{p 0}^{2}}\ {{p 3}^{2}}}+{{{p 0}^{2}}\ {{p 2}^{2}}}\right)}\ {{q 2}^{2}}}+{{\left({2 \ {{p 0}^{2}}\  p 1 \  p 2 \  q 1}+{{\left(-{6 \  p 0 \  p 2 \ {{p 3}^{2}}}-{2 \ {{p 0}^{3}}\  p 2}\right)}\  q 0}\right)}\  q 2}+{{\left({{{p 0}^{2}}\ {{p 3}^{2}}}+{{{p 0}^{2}}\ {{p 1}^{2}}}\right)}\ {{q 1}^{2}}}+{{\left(-{6 \  p 0 \  p 1 \ {{p 3}^{2}}}-{2 \ {{p 0}^{3}}\  p 1}\right)}\  q 0 \  q 1}+{{\left({{p 3}^{4}}+{{\left({{p 2}^{2}}+{{p 1}^{2}}+{4 \ {{p 0}^{2}}}\right)}\ {{p 3}^{2}}}+{{p 0}^{4}}\right)}\ {{q 0}^{2}}}\right)}\ {{q 3}^{2}}}+{{\left({2 \ {{p 0}^{2}}\  p 2 \  p 3 \ {{q 2}^{3}}}+{{\left({2 \ {{p 0}^{2}}\  p 1 \  p 3 \  q 1}+{{\left(-{6 \  p 0 \ {{p 2}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 3 \  q 0}\right)}\ {{q 2}^{2}}}+{{\left({2 \ {{p 0}^{2}}\  p 2 \  p 3 \ {{q 1}^{2}}}-{{1
2}\  p 0 \  p 1 \  p 2 \  p 3 \  q 0 \  q 1}+{{\left({2 \  p 2 \ {{p 3}^{3}}}+{{\left({2 \ {{p 2}^{3}}}+{{\left({2 \ {{p 1}^{2}}}+{8 \ {{p 0}^{2}}}\right)}\  p 2}\right)}\  p 3}\right)}\ {{q 0}^{2}}}\right)}\  q 2}+{2 \ {{p 0}^{2}}\  p 1 \  p 3 \ {{q 1}^{3}}}+{{\left(-{6 \  p 0 \ {{p 1}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 3 \  q 0 \ {{q 1}^{2}}}+{{\left({2 \  p 1 \ {{p 3}^{3}}}+{{\left({2 \  p 1 \ {{p 2}^{2}}}+{2 \ {{p 1}^{3}}}+{8 \ {{p 0}^{2}}\  p 1}\right)}\  p 3}\right)}\ {{q 0}^{2}}\  q 1}+{{\left(-{2 \  p 0 \ {{p 3}^{3}}}+{{\left(-{2 \  p 0 \ {{p 2}^{2}}}-{2 \  p 0 \ {{p 1}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 3}\right)}\ {{q 0}^{3}}}\right)}\  q 3}+{{{p 0}^{2}}\ {{p 2}^{2}}\ {{q 2}^{4}}}+{{\left({2 \ {{p 0}^{2}}\  p 1 \  p 2 \  q 1}+{{\left(-{2 \  p 0 \ {{p 2}^{3}}}-{2 \ {{p 0}^{3}}\  p 2}\right)}\  q 0}\right)}\ {{q 2}^{3}}}+{{\left({{\left({{{p 0}^{2}}\ {{p 2}^{2}}}+{{{p 0}^{2}}\ {{p 1}^{2}}}\right)}\ {{q 1}^{2}}}+{{\left(-{6 \  p 0 \  p 1 \ {{p 2}^{2}}}-{2 \ {{p 0}^{3}}\  p 1}\right)}\  q 0 \  q 1}+{{\left({{{p 2}^{2}}\ {{p 3}^{2}}}+{{p 2}^{4}}+{{\left({{p 1}^{2}}+{4 \ {{p 0}^{2}}}\right)}\ {{p 2}^{2}}}+{{p 0}^{4}}\right)}\ {{q 0}^{2}}}\right)}\ {{q 2}^{2}}}+{{\left({2 \ {{p 0}^{2}}\  p 1 \  p 2 \ {{q 1}^{3}}}+{{\left(-{6 \  p 0 \ {{p 1}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 2 \  q 0 \ {{q 1}^{2}}}+{{\left({2 \  p 1 \  p 2 \ {{p 3}^{2}}}+{2 \  p 1 \ {{p 2}^{3}}}+{{\left({2 \ {{p 1}^{3}}}+{8 \ {{p 0}^{2}}\  p 1}\right)}\  p 2}\right)}\ {{q 0}^{2}}\  q 1}+{{\left(-{2 \  p 0 \  p 2 \ {{p 3}^{2}}}-{2 \  p 0 \ {{p 2}^{3}}}+{{\left(-{2 \  p 0 \ {{p 1}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 2}\right)}\ {{q 0}^{3}}}\right)}\  q 2}+{{{p 0}^{2}}\ {{p 1}^{2}}\ {{q 1}^{4}}}+{{\left(-{2 \  p 0 \ {{p 1}^{3}}}-{2 \ {{p 0}^{3}}\  p 1}\right)}\  q 0 \ {{q 1}^{3}}}+{{\left({{{p 1}^{2}}\ {{p 3}^{2}}}+{{{p 1}^{2}}\ {{p 2}^{2}}}+{{p 1}^{4}}+{4 \ {{p 0}^{2}}\ {{p 1}^{2}}}+{{p 0}^{4}}\right)}\ {{q 0}^{2}}\ {{q 1}^{2}}}+{{\left(-{2 \  p 0 \  p 1 \ {{p 3}^{2}}}-{2 \  p 0 \  p 1 \ {{p 2}^{2}}}-{2 \  p 0 \ {{p 1}^{3}}}-{2 \ {{p 0}^{3}}\  p 1}\right)}\ {{q 0}^{3}}\  q 1}+{{\left({{{p 0}^{2}}\ {{p 3}^{2}}}+{{{p 0}^{2}}\ {{p 2}^{2}}}+{{{p 0}^{2}}\ {{p 1}^{2}}}\right)}\ {{q 0}^{4}}}}&{-{p 0 \  p 1 \ {{p 3}^{2}}\ {{q 3}^{4}}}+{{\left(-{2 \  p 0 \  p 1 \  p 2 \  p 3 \  q 2}+{{\left({p 0 \ {{p 3}^{3}}}-{2 \  p 0 \ {{p 1}^{2}}\  p 3}\right)}\  q 1}+{{\left({p 1 \ {{p 3}^{3}}}+{2 \ {{p 0}^{2}}\  p 1 \  p 3}\right)}\  q 0}\right)}\ {{q 3}^{3}}}+{{\left({{\left(-{p 0 \  p 1 \ {{p 3}^{2}}}-{p 0 \  p 1 \ {{p 2}^{2}}}\right)}\ {{q 2}^{2}}}+{{\left({{\left({3 \  p 0 \  p 2 \ {{p 3}^{2}}}-{2 \  p 0 \ {{p 1}^{2}}\  p 2}\right)}\  q 1}+{{\left({3 \  p 1 \  p 2 \ {{p 3}^{2}}}+{2 \ {{p 0}^{2}}\  p 1 \  p 2}\right)}\  q 0}\right)}\  q 2}+{{\left({2 \  p 0 \  p 1 \ {{p 3}^{2}}}-{p 0 \ {{p 1}^{3}}}\right)}\ {{q 1}^{2}}}+{{\left(-{{p 3}^{4}}+{{\left(-{{p 2}^{2}}+{2 \ {{p 1}^{2}}}-{2 \ {{p 0}^{2}}}\right)}\ {{p 3}^{2}}}+{2 \ {{p 0}^{2}}\ {{p 1}^{2}}}\right)}\  q 0 \  q 1}+{{\left(-{2 \  p 0 \  p 1 \ {{p 3}^{2}}}-{{{p 0}^{3}}\  p 1}\right)}\ {{q 0}^{2}}}\right)}\ {{q 3}^{2}}}+{{\left(-{2 \  p 0 \  p 1 \  p 2 \  p 3 \ {{q 2}^{3}}}+{{\left({{\left({3 \  p 0 \ {{p 2}^{2}}}-{2 \  p 0 \ {{p 1}^{2}}}\right)}\  p 3 \  q 1}+{{\left({3 \  p 1 \ {{p 2}^{2}}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 3 \  q 0}\right)}\ {{q 2}^{2}}}+{{\left({4 \  p 0 \  p 1 \  p 2 \  p 3 \ {{q 1}^{2}}}+{{\left(-{2 \  p 2 \ {{p 3}^{3}}}+{{\left(-{2 \ {{p 2}^{3}}}+{{\left({4 \ {{p 1}^{2}}}-{4 \ {{p 0}^{2}}}\right)}\  p 2}\right)}\  p 3}\right)}\  q 0 \  q 1}-{4 \  p 0 \  p 1 \  p 2 \  p 3 \ {{q 0}^{2}}}\right)}\  q 2}+{p 0 \ {{p 1}^{2}}\  p 3 \ {{q 1}^{3}}}+{{\left(-{2 \  p 1 \ {{p 3}^{3}}}+{{\left(-{2 \  p 1 \ {{p 2}^{2}}}+{{p 1}^{3}}-{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 3}\right)}\  q 0 \ {{q 1}^{2}}}+{{\left({2 \  p 0 \ {{p 3}^{3}}}+{{\left({2 \  p 0 \ {{p 2}^{2}}}-{2 \  p 0 \ {{p 1}^{2}}}+{{p 0}^{3}}\right)}\  p 3}\right)}\ {{q 0}^{2}}\  q 1}+{{{p 0}^{2}}\  p 1 \  p 3 \ {{q 0}^{3}}}\right)}\  q 3}-{p 0 \  p 1 \ {{p 2}^{2}}\ {{q 2}^{4}}}+{{\left({{\left({p 0 \ {{p 2}^{3}}}-{2 \  p 0 \ {{p 1}^{2}}\  p 2}\right)}\  q 1}+{{\left({p 1 \ {{p 2}^{3}}}+{2 \ {{p 0}^{2}}\  p 1 \  p 2}\right)}\  q 0}\right)}\ {{q 2}^{3}}}+{{\left({{\left({2 \  p 0 \  p 1 \ {{p 2}^{2}}}-{p 0 \ {{p 1}^{3}}}\right)}\ {{q 1}^{2}}}+{{\left(-{{{p 2}^{2}}\ {{p 3}^{2}}}-{{p 2}^{4}}+{{\left({2 \ {{p 1}^{2}}}-{2 \ {{p 0}^{2}}}\right)}\ {{p 2}^{2}}}+{2 \ {{p 0}^{2}}\ {{p 1}^{2}}}\right)}\  q 0 \  q 1}+{{\left(-{2 \  p 0 \  p 1 \ {{p 2}^{2}}}-{{{p 0}^{3}}\  p 1}\right)}\ {{q 0}^{2}}}\right)}\ {{q 2}^{2}}}+{{\left({p 0 \ {{p 1}^{2}}\  p 2 \ {{q 1}^{3}}}+{{\left(-{2 \  p 1 \  p 2 \ {{p 3}^{2}}}-{2 \  p 1 \ {{p 2}^{3}}}+{{\left({{p 1}^{3}}-{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 2}\right)}\  q 0 \ {{q 1}^{2}}}+{{\left({2 \  p 0 \  p 2 \ {{p 3}^{2}}}+{2 \  p 0 \ {{p 2}^{3}}}+{{\left(-{2 \  p 0 \ {{p 1}^{2}}}+{{p 0}^{3}}\right)}\  p 2}\right)}\ {{q 0}^{2}}\  q 1}+{{{p 0}^{2}}\  p 1 \  p 2 \ {{q 0}^{3}}}\right)}\  q 2}+{{\left(-{{{p 1}^{2}}\ {{p 3}^{2}}}-{{{p 1}^{2}}\ {{p 2}^{2}}}\right)}\  q 0 \ {{q 1}^{3}}}+{{\left({2 \  p 0 \  p 1 \ {{p 3}^{2}}}+{2 \  p 0 \  p 1 \ {{p 2}^{2}}}\right)}\ {{q 0}^{2}}\ {{q 1}^{2}}}+{{\left(-{{{p 0}^{2}}\ {{p 3}^{2}}}-{{{p 0}^{2}}\ {{p 2}^{2}}}\right)}\ {{q 0}^{3}}\  q 1}}&{-{p 0 \  p 2 \ {{p 3}^{2}}\ {{q 3}^{4}}}+{{\left({{\left({p 0 \ {{p 3}^{3}}}-{2 \  p 0 \ {{p 2}^{2}}\  p 3}\right)}\  q 2}-{2 \  p 0 \  p 1 \  p 2 \  p 3 \  q 1}+{{\left({p 2 \ {{p 3}^{3}}}+{2 \ {{p 0}^{2}}\  p 2 \  p 3}\right)}\  q 0}\right)}\ {{q 3}^{3}}}+{{\left({{\left({2 \  p 0 \  p 2 \ {{p 3}^{2}}}-{p 0 \ {{p 2}^{3}}}\right)}\ {{q 2}^{2}}}+{{\left({{\left({3 \  p 0 \  p 1 \ {{p 3}^{2}}}-{2 \  p 0 \  p 1 \ {{p 2}^{2}}}\right)}\  q 1}+{{\left(-{{p 3}^{4}}+{{\left({2 \ {{p 2}^{2}}}-{{p 1}^{2}}-{2 \ {{p 0}^{2}}}\right)}\ {{p 3}^{2}}}+{2 \ {{p 0}^{2}}\ {{p 2}^{2}}}\right)}\  q 0}\right)}\  q 2}+{{\left(-{p 0 \  p 2 \ {{p 3}^{2}}}-{p 0 \ {{p 1}^{2}}\  p 2}\right)}\ {{q 1}^{2}}}+{{\left({3 \  p 1 \  p 2 \ {{p 3}^{2}}}+{2 \ {{p 0}^{2}}\  p 1 \  p 2}\right)}\  q 0 \  q 1}+{{\left(-{2 \  p 0 \  p 2 \ {{p 3}^{2}}}-{{{p 0}^{3}}\  p 2}\right)}\ {{q 0}^{2}}}\right)}\ {{q 3}^{2}}}+{{\left({p 0 \ {{p 2}^{2}}\  p 3 \ {{q 2}^{3}}}+{{\left({4 \  p 0 \  p 1 \  p 2 \  p 3 \  q 1}+{{\left(-{2 \  p 2 \ {{p 3}^{3}}}+{{\left({{p 2}^{3}}+{{\left(-{2 \ {{p 1}^{2}}}-{2 \ {{p 0}^{2}}}\right)}\  p 2}\right)}\  p 3}\right)}\  q 0}\right)}\ {{q 2}^{2}}}+{{\left({{\left(-{2 \  p 0 \ {{p 2}^{2}}}+{3 \  p 0 \ {{p 1}^{2}}}\right)}\  p 3 \ {{q 1}^{2}}}+{{\left(-{2 \  p 1 \ {{p 3}^{3}}}+{{\left({4 \  p 1 \ {{p 2}^{2}}}-{2 \ {{p 1}^{3}}}-{4 \ {{p 0}^{2}}\  p 1}\right)}\  p 3}\right)}\  q 0 \  q 1}+{{\left({2 \  p 0 \ {{p 3}^{3}}}+{{\left(-{2 \  p 0 \ {{p 2}^{2}}}+{2 \  p 0 \ {{p 1}^{2}}}+{{p 0}^{3}}\right)}\  p 3}\right)}\ {{q 0}^{2}}}\right)}\  q 2}-{2 \  p 0 \  p 1 \  p 2 \  p 3 \ {{q 1}^{3}}}+{{\left({3 \ {{p 1}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\  p 2 \  p 3 \  q 0 \ {{q 1}^{2}}}-{4 \  p 0 \  p 1 \  p 2 \  p 3 \ {{q 0}^{2}}\  q 1}+{{{p 0}^{2}}\  p 2 \  p 3 \ {{q 0}^{3}}}\right)}\  q 3}+{{\left({p 0 \  p 1 \ {{p 2}^{2}}\  q 1}+{{\left(-{{{p 2}^{2}}\ {{p 3}^{2}}}-{{{p 1}^{2}}\ {{p 2}^{2}}}\right)}\  q 0}\right)}\ {{q 2}^{3}}}+{{\left({{\left(-{p 0 \ {{p 2}^{3}}}+{2 \  p 0 \ {{p 1}^{2}}\  p 2}\right)}\ {{q 1}^{2}}}+{{\left(-{2 \  p 1 \  p 2 \ {{p 3}^{2}}}+{p 1 \ {{p 2}^{3}}}+{{\left(-{2 \ {{p 1}^{3}}}-{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 2}\right)}\  q 0 \  q 1}+{{\left({2 \  p 0 \  p 2 \ {{p 3}^{2}}}+{2 \  p 0 \ {{p 1}^{2}}\  p 2}\right)}\ {{q 0}^{2}}}\right)}\ {{q 2}^{2}}}+{{\left({{\left(-{2 \  p 0 \  p 1 \ {{p 2}^{2}}}+{p 0 \ {{p 1}^{3}}}\right)}\ {{q 1}^{3}}}+{{\left(-{{{p 1}^{2}}\ {{p 3}^{2}}}+{{\left({2 \ {{p 1}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\ {{p 2}^{2}}}-{{p 1}^{4}}-{2 \ {{p 0}^{2}}\ {{p 1}^{2}}}\right)}\  q 0 \ {{q 1}^{2}}}+{{\left({2 \  p 0 \  p 1 \ {{p 3}^{2}}}-{2 \  p 0 \  p 1 \ {{p 2}^{2}}}+{2 \  p 0 \ {{p 1}^{3}}}+{{{p 0}^{3}}\  p 1}\right)}\ {{q 0}^{2}}\  q 1}+{{\left(-{{{p 0}^{2}}\ {{p 3}^{2}}}-{{{p 0}^{2}}\ {{p 1}^{2}}}\right)}\ {{q 0}^{3}}}\right)}\  q 2}-{p 0 \ {{p 1}^{2}}\  p 2 \ {{q 1}^{4}}}+{{\left({{p 1}^{3}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 2 \  q 0 \ {{q 1}^{3}}}+{{\left(-{2 \  p 0 \ {{p 1}^{2}}}-{{p 0}^{3}}\right)}\  p 2 \ {{q 0}^{2}}\ {{q 1}^{2}}}+{{{p 0}^{2}}\  p 1 \  p 2 \ {{q 0}^{3}}\  q 1}}&{{{\left({p 0 \  p 2 \ {{p 3}^{2}}\  q 2}+{p 0 \  p 1 \ {{p 3}^{2}}\  q 1}+{{\left(-{{p 2}^{2}}-{{p 1}^{2}}\right)}\ {{p 3}^{2}}\  q 0}\right)}\ {{q 3}^{3}}}+{{\left({{\left(-{p 0 \ {{p 3}^{3}}}+{2 \  p 0 \ {{p 2}^{2}}\  p 3}\right)}\ {{q 2}^{2}}}+{{\left({4 \  p 0 \  p 1 \  p 2 \  p 3 \  q 1}+{{\left({p 2 \ {{p 3}^{3}}}+{{\left(-{2 \ {{p 2}^{3}}}+{{\left(-{2 \ {{p 1}^{2}}}-{2 \ {{p 0}^{2}}}\right)}\  p 2}\right)}\  p 3}\right)}\  q 0}\right)}\  q 2}+{{\left(-{p 0 \ {{p 3}^{3}}}+{2 \  p 0 \ {{p 1}^{2}}\  p 3}\right)}\ {{q 1}^{2}}}+{{\left({p 1 \ {{p 3}^{3}}}+{{\left(-{2 \  p 1 \ {{p 2}^{2}}}-{2 \ {{p 1}^{3}}}-{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 3}\right)}\  q 0 \  q 1}+{{\left({2 \  p 0 \ {{p 2}^{2}}}+{2 \  p 0 \ {{p 1}^{2}}}\right)}\  p 3 \ {{q 0}^{2}}}\right)}\ {{q 3}^{2}}}+{{\left({{\left(-{2 \  p 0 \  p 2 \ {{p 3}^{2}}}+{p 0 \ {{p 2}^{3}}}\right)}\ {{q 2}^{3}}}+{{\left({{\left(-{2 \  p 0 \  p 1 \ {{p 3}^{2}}}+{3 \  p 0 \  p 1 \ {{p 2}^{2}}}\right)}\  q 1}+{{\left({{\left({2 \ {{p 2}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\ {{p 3}^{2}}}-{{p 2}^{4}}+{{\left(-{{p 1}^{2}}-{2 \ {{p 0}^{2}}}\right)}\ {{p 2}^{2}}}\right)}\  q 0}\right)}\ {{q 2}^{2}}}+{{\left({{\left(-{2 \  p 0 \  p 2 \ {{p 3}^{2}}}+{3 \  p 0 \ {{p 1}^{2}}\  p 2}\right)}\ {{q 1}^{2}}}+{{\left({4 \  p 1 \  p 2 \ {{p 3}^{2}}}-{2 \  p 1 \ {{p 2}^{3}}}+{{\left(-{2 \ {{p 1}^{3}}}-{4 \ {{p 0}^{2}}\  p 1}\right)}\  p 2}\right)}\  q 0 \  q 1}+{{\left(-{2 \  p 0 \  p 2 \ {{p 3}^{2}}}+{2 \  p 0 \ {{p 2}^{3}}}+{{\left({2 \  p 0 \ {{p 1}^{2}}}+{{p 0}^{3}}\right)}\  p 2}\right)}\ {{q 0}^{2}}}\right)}\  q 2}+{{\left(-{2 \  p 0 \  p 1 \ {{p 3}^{2}}}+{p 0 \ {{p 1}^{3}}}\right)}\ {{q 1}^{3}}}+{{\left({{\left({2 \ {{p 1}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\ {{p 3}^{2}}}-{{{p 1}^{2}}\ {{p 2}^{2}}}-{{p 1}^{4}}-{2 \ {{p 0}^{2}}\ {{p 1}^{2}}}\right)}\  q 0 \ {{q 1}^{2}}}+{{\left(-{2 \  p 0 \  p 1 \ {{p 3}^{2}}}+{2 \  p 0 \  p 1 \ {{p 2}^{2}}}+{2 \  p 0 \ {{p 1}^{3}}}+{{{p 0}^{3}}\  p 1}\right)}\ {{q 0}^{2}}\  q 1}+{{\left(-{{{p 0}^{2}}\ {{p 2}^{2}}}-{{{p 0}^{2}}\ {{p 1}^{2}}}\right)}\ {{q 0}^{3}}}\right)}\  q 3}-{p 0 \ {{p 2}^{2}}\  p 3 \ {{q 2}^{4}}}+{{\left(-{2 \  p 0 \  p 1 \  p 2 \  p 3 \  q 1}+{{\left({{p 2}^{3}}+{2 \ {{p 0}^{2}}\  p 2}\right)}\  p 3 \  q 0}\right)}\ {{q 2}^{3}}}+{{\left({{\left(-{p 0 \ {{p 2}^{2}}}-{p 0 \ {{p 1}^{2}}}\right)}\  p 3 \ {{q 1}^{2}}}+{{\left({3 \  p 1 \ {{p 2}^{2}}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 3 \  q 0 \  q 1}+{{\left(-{2 \  p 0 \ {{p 2}^{2}}}-{{p 0}^{3}}\right)}\  p 3 \ {{q 0}^{2}}}\right)}\ {{q 2}^{2}}}+{{\left(-{2 \  p 0 \  p 1 \  p 2 \  p 3 \ {{q 1}^{3}}}+{{\left({3 \ {{p 1}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\  p 2 \  p 3 \  q 0 \ {{q 1}^{2}}}-{4 \  p 0 \  p 1 \  p 2 \  p 3 \ {{q 0}^{2}}\  q 1}+{{{p 0}^{2}}\  p 2 \  p 3 \ {{q 0}^{3}}}\right)}\  q 2}-{p 0 \ {{p 1}^{2}}\  p 3 \ {{q 1}^{4}}}+{{\left({{p 1}^{3}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 3 \  q 0 \ {{q 1}^{3}}}+{{\left(-{2 \  p 0 \ {{p 1}^{2}}}-{{p 0}^{3}}\right)}\  p 3 \ {{q 0}^{2}}\ {{q 1}^{2}}}+{{{p 0}^{2}}\  p 1 \  p 3 \ {{q 0}^{3}}\  q 1}}
\
{{p 0 \  p 1 \ {{p 3}^{2}}\ {{q 3}^{4}}}+{{\left({2 \  p 0 \  p 1 \  p 2 \  p 3 \  q 2}+{{\left(-{p 0 \ {{p 3}^{3}}}+{2 \  p 0 \ {{p 1}^{2}}\  p 3}\right)}\  q 1}+{{\left(-{p 1 \ {{p 3}^{3}}}-{2 \ {{p 0}^{2}}\  p 1 \  p 3}\right)}\  q 0}\right)}\ {{q 3}^{3}}}+{{\left({{\left({p 0 \  p 1 \ {{p 3}^{2}}}+{p 0 \  p 1 \ {{p 2}^{2}}}\right)}\ {{q 2}^{2}}}+{{\left({{\left(-{3 \  p 0 \  p 2 \ {{p 3}^{2}}}+{2 \  p 0 \ {{p 1}^{2}}\  p 2}\right)}\  q 1}+{{\left(-{3 \  p 1 \  p 2 \ {{p 3}^{2}}}-{2 \ {{p 0}^{2}}\  p 1 \  p 2}\right)}\  q 0}\right)}\  q 2}+{{\left(-{2 \  p 0 \  p 1 \ {{p 3}^{2}}}+{p 0 \ {{p 1}^{3}}}\right)}\ {{q 1}^{2}}}+{{\left({{p 3}^{4}}+{{\left({{p 2}^{2}}-{2 \ {{p 1}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\ {{p 3}^{2}}}-{2 \ {{p 0}^{2}}\ {{p 1}^{2}}}\right)}\  q 0 \  q 1}+{{\left({2 \  p 0 \  p 1 \ {{p 3}^{2}}}+{{{p 0}^{3}}\  p 1}\right)}\ {{q 0}^{2}}}\right)}\ {{q 3}^{2}}}+{{\left({2 \  p 0 \  p 1 \  p 2 \  p 3 \ {{q 2}^{3}}}+{{\left({{\left(-{3 \  p 0 \ {{p 2}^{2}}}+{2 \  p 0 \ {{p 1}^{2}}}\right)}\  p 3 \  q 1}+{{\left(-{3 \  p 1 \ {{p 2}^{2}}}-{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 3 \  q 0}\right)}\ {{q 2}^{2}}}+{{\left(-{4 \  p 0 \  p 1 \  p 2 \  p 3 \ {{q 1}^{2}}}+{{\left({2 \  p 2 \ {{p 3}^{3}}}+{{\left({2 \ {{p 2}^{3}}}+{{\left(-{4 \ {{p 1}^{2}}}+{4 \ {{p 0}^{2}}}\right)}\  p 2}\right)}\  p 3}\right)}\  q 0 \  q 1}+{4 \  p 0 \  p 1 \  p 2 \  p 3 \ {{q 0}^{2}}}\right)}\  q 2}-{p 0 \ {{p 1}^{2}}\  p 3 \ {{q 1}^{3}}}+{{\left({2 \  p 1 \ {{p 3}^{3}}}+{{\left({2 \  p 1 \ {{p 2}^{2}}}-{{p 1}^{3}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 3}\right)}\  q 0 \ {{q 1}^{2}}}+{{\left(-{2 \  p 0 \ {{p 3}^{3}}}+{{\left(-{2 \  p 0 \ {{p 2}^{2}}}+{2 \  p 0 \ {{p 1}^{2}}}-{{p 0}^{3}}\right)}\  p 3}\right)}\ {{q 0}^{2}}\  q 1}-{{{p 0}^{2}}\  p 1 \  p 3 \ {{q 0}^{3}}}\right)}\  q 3}+{p 0 \  p 1 \ {{p 2}^{2}}\ {{q 2}^{4}}}+{{\left({{\left(-{p 0 \ {{p 2}^{3}}}+{2 \  p 0 \ {{p 1}^{2}}\  p 2}\right)}\  q 1}+{{\left(-{p 1 \ {{p 2}^{3}}}-{2 \ {{p 0}^{2}}\  p 1 \  p 2}\right)}\  q 0}\right)}\ {{q 2}^{3}}}+{{\left({{\left(-{2 \  p 0 \  p 1 \ {{p 2}^{2}}}+{p 0 \ {{p 1}^{3}}}\right)}\ {{q 1}^{2}}}+{{\left({{{p 2}^{2}}\ {{p 3}^{2}}}+{{p 2}^{4}}+{{\left(-{2 \ {{p 1}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\ {{p 2}^{2}}}-{2 \ {{p 0}^{2}}\ {{p 1}^{2}}}\right)}\  q 0 \  q 1}+{{\left({2 \  p 0 \  p 1 \ {{p 2}^{2}}}+{{{p 0}^{3}}\  p 1}\right)}\ {{q 0}^{2}}}\right)}\ {{q 2}^{2}}}+{{\left(-{p 0 \ {{p 1}^{2}}\  p 2 \ {{q 1}^{3}}}+{{\left({2 \  p 1 \  p 2 \ {{p 3}^{2}}}+{2 \  p 1 \ {{p 2}^{3}}}+{{\left(-{{p 1}^{3}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 2}\right)}\  q 0 \ {{q 1}^{2}}}+{{\left(-{2 \  p 0 \  p 2 \ {{p 3}^{2}}}-{2 \  p 0 \ {{p 2}^{3}}}+{{\left({2 \  p 0 \ {{p 1}^{2}}}-{{p 0}^{3}}\right)}\  p 2}\right)}\ {{q 0}^{2}}\  q 1}-{{{p 0}^{2}}\  p 1 \  p 2 \ {{q 0}^{3}}}\right)}\  q 2}+{{\left({{{p 1}^{2}}\ {{p 3}^{2}}}+{{{p 1}^{2}}\ {{p 2}^{2}}}\right)}\  q 0 \ {{q 1}^{3}}}+{{\left(-{2 \  p 0 \  p 1 \ {{p 3}^{2}}}-{2 \  p 0 \  p 1 \ {{p 2}^{2}}}\right)}\ {{q 0}^{2}}\ {{q 1}^{2}}}+{{\left({{{p 0}^{2}}\ {{p 3}^{2}}}+{{{p 0}^{2}}\ {{p 2}^{2}}}\right)}\ {{q 0}^{3}}\  q 1}}&{-{{{p 1}^{2}}\ {{p 3}^{2}}\ {{q 3}^{4}}}+{{\left(-{2 \ {{p 1}^{2}}\  p 2 \  p 3 \  q 2}+{{\left({2 \  p 1 \ {{p 3}^{3}}}-{2 \ {{p 1}^{3}}\  p 3}\right)}\  q 1}+{2 \  p 0 \ {{p 1}^{2}}\  p 3 \  q 0}\right)}\ {{q 3}^{3}}}+{{\left({{\left(-{{{p 1}^{2}}\ {{p 3}^{2}}}-{{{p 1}^{2}}\ {{p 2}^{2}}}\right)}\ {{q 2}^{2}}}+{{\left({{\left({6 \  p 1 \  p 2 \ {{p 3}^{2}}}-{2 \ {{p 1}^{3}}\  p 2}\right)}\  q 1}+{2 \  p 0 \ {{p 1}^{2}}\  p 2 \  q 0}\right)}\  q 2}+{{\left(-{{p 3}^{4}}+{{\left(-{{p 2}^{2}}+{4 \ {{p 1}^{2}}}+{{p 0}^{2}}\right)}\ {{p 3}^{2}}}-{{p 1}^{4}}\right)}\ {{q 1}^{2}}}+{{\left(-{6 \  p 0 \  p 1 \ {{p 3}^{2}}}+{2 \  p 0 \ {{p 1}^{3}}}\right)}\  q 0 \  q 1}+{{\left({{{p 1}^{2}}\ {{p 3}^{2}}}-{{{p 0}^{2}}\ {{p 1}^{2}}}\right)}\ {{q 0}^{2}}}\right)}\ {{q 3}^{2}}}+{{\left(-{2 \ {{p 1}^{2}}\  p 2 \  p 3 \ {{q 2}^{3}}}+{{\left({{\left({6 \  p 1 \ {{p 2}^{2}}}-{2 \ {{p 1}^{3}}}\right)}\  p 3 \  q 1}+{2 \  p 0 \ {{p 1}^{2}}\  p 3 \  q 0}\right)}\ {{q 2}^{2}}}+{{\left({{\left(-{2 \  p 2 \ {{p 3}^{3}}}+{{\left(-{2 \ {{p 2}^{3}}}+{{\left({8 \ {{p 1}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\  p 2}\right)}\  p 3}\right)}\ {{q 1}^{2}}}-{{12}\  p 0 \  p 1 \  p 2 \  p 3 \  q 0 \  q 1}+{2 \ {{p 1}^{2}}\  p 2 \  p 3 \ {{q 0}^{2}}}\right)}\  q 2}+{{\left(-{2 \  p 1 \ {{p 3}^{3}}}+{{\left(-{2 \  p 1 \ {{p 2}^{2}}}+{2 \ {{p 1}^{3}}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 3}\right)}\ {{q 1}^{3}}}+{{\left({2 \  p 0 \ {{p 3}^{3}}}+{{\left({2 \  p 0 \ {{p 2}^{2}}}-{8 \  p 0 \ {{p 1}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 3}\right)}\  q 0 \ {{q 1}^{2}}}+{{\left({2 \ {{p 1}^{3}}}+{6 \ {{p 0}^{2}}\  p 1}\right)}\  p 3 \ {{q 0}^{2}}\  q 1}-{2 \  p 0 \ {{p 1}^{2}}\  p 3 \ {{q 0}^{3}}}\right)}\  q 3}-{{{p 1}^{2}}\ {{p 2}^{2}}\ {{q 2}^{4}}}+{{\left({{\left({2 \  p 1 \ {{p 2}^{3}}}-{2 \ {{p 1}^{3}}\  p 2}\right)}\  q 1}+{2 \  p 0 \ {{p 1}^{2}}\  p 2 \  q 0}\right)}\ {{q 2}^{3}}}+{{\left({{\left(-{{{p 2}^{2}}\ {{p 3}^{2}}}-{{p 2}^{4}}+{{\left({4 \ {{p 1}^{2}}}+{{p 0}^{2}}\right)}\ {{p 2}^{2}}}-{{p 1}^{4}}\right)}\ {{q 1}^{2}}}+{{\left(-{6 \  p 0 \  p 1 \ {{p 2}^{2}}}+{2 \  p 0 \ {{p 1}^{3}}}\right)}\  q 0 \  q 1}+{{\left({{{p 1}^{2}}\ {{p 2}^{2}}}-{{{p 0}^{2}}\ {{p 1}^{2}}}\right)}\ {{q 0}^{2}}}\right)}\ {{q 2}^{2}}}+{{\left({{\left(-{2 \  p 1 \  p 2 \ {{p 3}^{2}}}-{2 \  p 1 \ {{p 2}^{3}}}+{{\left({2 \ {{p 1}^{3}}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 2}\right)}\ {{q 1}^{3}}}+{{\left({2 \  p 0 \  p 2 \ {{p 3}^{2}}}+{2 \  p 0 \ {{p 2}^{3}}}+{{\left(-{8 \  p 0 \ {{p 1}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 2}\right)}\  q 0 \ {{q 1}^{2}}}+{{\left({2 \ {{p 1}^{3}}}+{6 \ {{p 0}^{2}}\  p 1}\right)}\  p 2 \ {{q 0}^{2}}\  q 1}-{2 \  p 0 \ {{p 1}^{2}}\  p 2 \ {{q 0}^{3}}}\right)}\  q 2}+{{\left(-{{{p 1}^{2}}\ {{p 3}^{2}}}-{{{p 1}^{2}}\ {{p 2}^{2}}}+{{{p 0}^{2}}\ {{p 1}^{2}}}\right)}\ {{q 1}^{4}}}+{{\left({2 \  p 0 \  p 1 \ {{p 3}^{2}}}+{2 \  p 0 \  p 1 \ {{p 2}^{2}}}-{2 \  p 0 \ {{p 1}^{3}}}-{2 \ {{p 0}^{3}}\  p 1}\right)}\  q 0 \ {{q 1}^{3}}}+{{\left(-{{{p 0}^{2}}\ {{p 3}^{2}}}-{{{p 0}^{2}}\ {{p 2}^{2}}}+{{p 1}^{4}}+{4 \ {{p 0}^{2}}\ {{p 1}^{2}}}+{{p 0}^{4}}\right)}\ {{q 0}^{2}}\ {{q 1}^{2}}}+{{\left(-{2 \  p 0 \ {{p 1}^{3}}}-{2 \ {{p 0}^{3}}\  p 1}\right)}\ {{q 0}^{3}}\  q 1}+{{{p 0}^{2}}\ {{p 1}^{2}}\ {{q 0}^{4}}}}&{-{p 1 \  p 2 \ {{p 3}^{2}}\ {{q 3}^{4}}}+{{\left({{\left({p 1 \ {{p 3}^{3}}}-{2 \  p 1 \ {{p 2}^{2}}\  p 3}\right)}\  q 2}+{{\left({p 2 \ {{p 3}^{3}}}-{2 \ {{p 1}^{2}}\  p 2 \  p 3}\right)}\  q 1}+{2 \  p 0 \  p 1 \  p 2 \  p 3 \  q 0}\right)}\ {{q 3}^{3}}}+{{\left({{\left({2 \  p 1 \  p 2 \ {{p 3}^{2}}}-{p 1 \ {{p 2}^{3}}}\right)}\ {{q 2}^{2}}}+{{\left({{\left(-{{p 3}^{4}}+{{\left({2 \ {{p 2}^{2}}}+{2 \ {{p 1}^{2}}}+{{p 0}^{2}}\right)}\ {{p 3}^{2}}}-{2 \ {{p 1}^{2}}\ {{p 2}^{2}}}\right)}\  q 1}+{{\left(-{3 \  p 0 \  p 1 \ {{p 3}^{2}}}+{2 \  p 0 \  p 1 \ {{p 2}^{2}}}\right)}\  q 0}\right)}\  q 2}+{{\left({2 \  p 1 \  p 2 \ {{p 3}^{2}}}-{{{p 1}^{3}}\  p 2}\right)}\ {{q 1}^{2}}}+{{\left(-{3 \  p 0 \  p 2 \ {{p 3}^{2}}}+{2 \  p 0 \ {{p 1}^{2}}\  p 2}\right)}\  q 0 \  q 1}+{{\left({p 1 \  p 2 \ {{p 3}^{2}}}-{{{p 0}^{2}}\  p 1 \  p 2}\right)}\ {{q 0}^{2}}}\right)}\ {{q 3}^{2}}}+{{\left({p 1 \ {{p 2}^{2}}\  p 3 \ {{q 2}^{3}}}+{{\left({{\left(-{2 \  p 2 \ {{p 3}^{3}}}+{{\left({{p 2}^{3}}+{{\left({2 \ {{p 1}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\  p 2}\right)}\  p 3}\right)}\  q 1}-{4 \  p 0 \  p 1 \  p 2 \  p 3 \  q 0}\right)}\ {{q 2}^{2}}}+{{\left({{\left(-{2 \  p 1 \ {{p 3}^{3}}}+{{\left({2 \  p 1 \ {{p 2}^{2}}}+{{p 1}^{3}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 3}\right)}\ {{q 1}^{2}}}+{{\left({2 \  p 0 \ {{p 3}^{3}}}+{{\left(-{4 \  p 0 \ {{p 2}^{2}}}-{4 \  p 0 \ {{p 1}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 3}\right)}\  q 0 \  q 1}+{{\left({2 \  p 1 \ {{p 2}^{2}}}+{3 \ {{p 0}^{2}}\  p 1}\right)}\  p 3 \ {{q 0}^{2}}}\right)}\  q 2}+{{{p 1}^{2}}\  p 2 \  p 3 \ {{q 1}^{3}}}-{4 \  p 0 \  p 1 \  p 2 \  p 3 \  q 0 \ {{q 1}^{2}}}+{{\left({2 \ {{p 1}^{2}}}+{3 \ {{p 0}^{2}}}\right)}\  p 2 \  p 3 \ {{q 0}^{2}}\  q 1}-{2 \  p 0 \  p 1 \  p 2 \  p 3 \ {{q 0}^{3}}}\right)}\  q 3}+{{\left({{\left(-{{{p 2}^{2}}\ {{p 3}^{2}}}+{{{p 0}^{2}}\ {{p 2}^{2}}}\right)}\  q 1}-{p 0 \  p 1 \ {{p 2}^{2}}\  q 0}\right)}\ {{q 2}^{3}}}+{{\left({{\left(-{2 \  p 1 \  p 2 \ {{p 3}^{2}}}+{2 \ {{p 0}^{2}}\  p 1 \  p 2}\right)}\ {{q 1}^{2}}}+{{\left({2 \  p 0 \  p 2 \ {{p 3}^{2}}}-{p 0 \ {{p 2}^{3}}}+{{\left(-{2 \  p 0 \ {{p 1}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 2}\right)}\  q 0 \  q 1}+{{\left({p 1 \ {{p 2}^{3}}}+{2 \ {{p 0}^{2}}\  p 1 \  p 2}\right)}\ {{q 0}^{2}}}\right)}\ {{q 2}^{2}}}+{{\left({{\left(-{{{p 1}^{2}}\ {{p 3}^{2}}}+{{{p 0}^{2}}\ {{p 1}^{2}}}\right)}\ {{q 1}^{3}}}+{{\left({2 \  p 0 \  p 1 \ {{p 3}^{2}}}-{2 \  p 0 \  p 1 \ {{p 2}^{2}}}-{p 0 \ {{p 1}^{3}}}-{2 \ {{p 0}^{3}}\  p 1}\right)}\  q 0 \ {{q 1}^{2}}}+{{\left(-{{{p 0}^{2}}\ {{p 3}^{2}}}+{{\left({2 \ {{p 1}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\ {{p 2}^{2}}}+{2 \ {{p 0}^{2}}\ {{p 1}^{2}}}+{{p 0}^{4}}\right)}\ {{q 0}^{2}}\  q 1}+{{\left(-{2 \  p 0 \  p 1 \ {{p 2}^{2}}}-{{{p 0}^{3}}\  p 1}\right)}\ {{q 0}^{3}}}\right)}\  q 2}-{p 0 \ {{p 1}^{2}}\  p 2 \  q 0 \ {{q 1}^{3}}}+{{\left({{p 1}^{3}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 2 \ {{q 0}^{2}}\ {{q 1}^{2}}}+{{\left(-{2 \  p 0 \ {{p 1}^{2}}}-{{p 0}^{3}}\right)}\  p 2 \ {{q 0}^{3}}\  q 1}+{{{p 0}^{2}}\  p 1 \  p 2 \ {{q 0}^{4}}}}&{{{\left({p 1 \  p 2 \ {{p 3}^{2}}\  q 2}+{{\left(-{{p 2}^{2}}+{{p 0}^{2}}\right)}\ {{p 3}^{2}}\  q 1}-{p 0 \  p 1 \ {{p 3}^{2}}\  q 0}\right)}\ {{q 3}^{3}}}+{{\left({{\left(-{p 1 \ {{p 3}^{3}}}+{2 \  p 1 \ {{p 2}^{2}}\  p 3}\right)}\ {{q 2}^{2}}}+{{\left({{\left({p 2 \ {{p 3}^{3}}}+{{\left(-{2 \ {{p 2}^{3}}}+{{\left({2 \ {{p 1}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\  p 2}\right)}\  p 3}\right)}\  q 1}-{4 \  p 0 \  p 1 \  p 2 \  p 3 \  q 0}\right)}\  q 2}+{{\left(-{2 \  p 1 \ {{p 2}^{2}}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 3 \ {{q 1}^{2}}}+{{\left(-{p 0 \ {{p 3}^{3}}}+{{\left({2 \  p 0 \ {{p 2}^{2}}}-{2 \  p 0 \ {{p 1}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 3}\right)}\  q 0 \  q 1}+{{\left({p 1 \ {{p 3}^{3}}}+{2 \ {{p 0}^{2}}\  p 1 \  p 3}\right)}\ {{q 0}^{2}}}\right)}\ {{q 3}^{2}}}+{{\left({{\left(-{2 \  p 1 \  p 2 \ {{p 3}^{2}}}+{p 1 \ {{p 2}^{3}}}\right)}\ {{q 2}^{3}}}+{{\left({{\left({{\left({2 \ {{p 2}^{2}}}-{2 \ {{p 1}^{2}}}\right)}\ {{p 3}^{2}}}-{{p 2}^{4}}+{{\left({2 \ {{p 1}^{2}}}+{{p 0}^{2}}\right)}\ {{p 2}^{2}}}\right)}\  q 1}+{{\left({2 \  p 0 \  p 1 \ {{p 3}^{2}}}-{3 \  p 0 \  p 1 \ {{p 2}^{2}}}\right)}\  q 0}\right)}\ {{q 2}^{2}}}+{{\left({{\left({2 \  p 1 \  p 2 \ {{p 3}^{2}}}-{2 \  p 1 \ {{p 2}^{3}}}+{{\left({{p 1}^{3}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 2}\right)}\ {{q 1}^{2}}}+{{\left(-{4 \  p 0 \  p 2 \ {{p 3}^{2}}}+{2 \  p 0 \ {{p 2}^{3}}}+{{\left(-{4 \  p 0 \ {{p 1}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 2}\right)}\  q 0 \  q 1}+{{\left({2 \  p 1 \  p 2 \ {{p 3}^{2}}}+{3 \ {{p 0}^{2}}\  p 1 \  p 2}\right)}\ {{q 0}^{2}}}\right)}\  q 2}+{{\left(-{{{p 1}^{2}}\ {{p 2}^{2}}}+{{{p 0}^{2}}\ {{p 1}^{2}}}\right)}\ {{q 1}^{3}}}+{{\left(-{2 \  p 0 \  p 1 \ {{p 3}^{2}}}+{2 \  p 0 \  p 1 \ {{p 2}^{2}}}-{p 0 \ {{p 1}^{3}}}-{2 \ {{p 0}^{3}}\  p 1}\right)}\  q 0 \ {{q 1}^{2}}}+{{\left({{\left({2 \ {{p 1}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\ {{p 3}^{2}}}-{{{p 0}^{2}}\ {{p 2}^{2}}}+{2 \ {{p 0}^{2}}\ {{p 1}^{2}}}+{{p 0}^{4}}\right)}\ {{q 0}^{2}}\  q 1}+{{\left(-{2 \  p 0 \  p 1 \ {{p 3}^{2}}}-{{{p 0}^{3}}\  p 1}\right)}\ {{q 0}^{3}}}\right)}\  q 3}-{p 1 \ {{p 2}^{2}}\  p 3 \ {{q 2}^{4}}}+{{\left({{\left({{p 2}^{3}}-{2 \ {{p 1}^{2}}\  p 2}\right)}\  p 3 \  q 1}+{2 \  p 0 \  p 1 \  p 2 \  p 3 \  q 0}\right)}\ {{q 2}^{3}}}+{{\left({{\left({2 \  p 1 \ {{p 2}^{2}}}-{{p 1}^{3}}\right)}\  p 3 \ {{q 1}^{2}}}+{{\left(-{3 \  p 0 \ {{p 2}^{2}}}+{2 \  p 0 \ {{p 1}^{2}}}\right)}\  p 3 \  q 0 \  q 1}+{{\left({p 1 \ {{p 2}^{2}}}-{{{p 0}^{2}}\  p 1}\right)}\  p 3 \ {{q 0}^{2}}}\right)}\ {{q 2}^{2}}}+{{\left({{{p 1}^{2}}\  p 2 \  p 3 \ {{q 1}^{3}}}-{4 \  p 0 \  p 1 \  p 2 \  p 3 \  q 0 \ {{q 1}^{2}}}+{{\left({2 \ {{p 1}^{2}}}+{3 \ {{p 0}^{2}}}\right)}\  p 2 \  p 3 \ {{q 0}^{2}}\  q 1}-{2 \  p 0 \  p 1 \  p 2 \  p 3 \ {{q 0}^{3}}}\right)}\  q 2}-{p 0 \ {{p 1}^{2}}\  p 3 \  q 0 \ {{q 1}^{3}}}+{{\left({{p 1}^{3}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 3 \ {{q 0}^{2}}\ {{q 1}^{2}}}+{{\left(-{2 \  p 0 \ {{p 1}^{2}}}-{{p 0}^{3}}\right)}\  p 3 \ {{q 0}^{3}}\  q 1}+{{{p 0}^{2}}\  p 1 \  p 3 \ {{q 0}^{4}}}}
\
{{p 0 \  p 2 \ {{p 3}^{2}}\ {{q 3}^{4}}}+{{\left({{\left(-{p 0 \ {{p 3}^{3}}}+{2 \  p 0 \ {{p 2}^{2}}\  p 3}\right)}\  q 2}+{2 \  p 0 \  p 1 \  p 2 \  p 3 \  q 1}+{{\left(-{p 2 \ {{p 3}^{3}}}-{2 \ {{p 0}^{2}}\  p 2 \  p 3}\right)}\  q 0}\right)}\ {{q 3}^{3}}}+{{\left({{\left(-{2 \  p 0 \  p 2 \ {{p 3}^{2}}}+{p 0 \ {{p 2}^{3}}}\right)}\ {{q 2}^{2}}}+{{\left({{\left(-{3 \  p 0 \  p 1 \ {{p 3}^{2}}}+{2 \  p 0 \  p 1 \ {{p 2}^{2}}}\right)}\  q 1}+{{\left({{p 3}^{4}}+{{\left(-{2 \ {{p 2}^{2}}}+{{p 1}^{2}}+{2 \ {{p 0}^{2}}}\right)}\ {{p 3}^{2}}}-{2 \ {{p 0}^{2}}\ {{p 2}^{2}}}\right)}\  q 0}\right)}\  q 2}+{{\left({p 0 \  p 2 \ {{p 3}^{2}}}+{p 0 \ {{p 1}^{2}}\  p 2}\right)}\ {{q 1}^{2}}}+{{\left(-{3 \  p 1 \  p 2 \ {{p 3}^{2}}}-{2 \ {{p 0}^{2}}\  p 1 \  p 2}\right)}\  q 0 \  q 1}+{{\left({2 \  p 0 \  p 2 \ {{p 3}^{2}}}+{{{p 0}^{3}}\  p 2}\right)}\ {{q 0}^{2}}}\right)}\ {{q 3}^{2}}}+{{\left(-{p 0 \ {{p 2}^{2}}\  p 3 \ {{q 2}^{3}}}+{{\left(-{4 \  p 0 \  p 1 \  p 2 \  p 3 \  q 1}+{{\left({2 \  p 2 \ {{p 3}^{3}}}+{{\left(-{{p 2}^{3}}+{{\left({2 \ {{p 1}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\  p 2}\right)}\  p 3}\right)}\  q 0}\right)}\ {{q 2}^{2}}}+{{\left({{\left({2 \  p 0 \ {{p 2}^{2}}}-{3 \  p 0 \ {{p 1}^{2}}}\right)}\  p 3 \ {{q 1}^{2}}}+{{\left({2 \  p 1 \ {{p 3}^{3}}}+{{\left(-{4 \  p 1 \ {{p 2}^{2}}}+{2 \ {{p 1}^{3}}}+{4 \ {{p 0}^{2}}\  p 1}\right)}\  p 3}\right)}\  q 0 \  q 1}+{{\left(-{2 \  p 0 \ {{p 3}^{3}}}+{{\left({2 \  p 0 \ {{p 2}^{2}}}-{2 \  p 0 \ {{p 1}^{2}}}-{{p 0}^{3}}\right)}\  p 3}\right)}\ {{q 0}^{2}}}\right)}\  q 2}+{2 \  p 0 \  p 1 \  p 2 \  p 3 \ {{q 1}^{3}}}+{{\left(-{3 \ {{p 1}^{2}}}-{2 \ {{p 0}^{2}}}\right)}\  p 2 \  p 3 \  q 0 \ {{q 1}^{2}}}+{4 \  p 0 \  p 1 \  p 2 \  p 3 \ {{q 0}^{2}}\  q 1}-{{{p 0}^{2}}\  p 2 \  p 3 \ {{q 0}^{3}}}\right)}\  q 3}+{{\left(-{p 0 \  p 1 \ {{p 2}^{2}}\  q 1}+{{\left({{{p 2}^{2}}\ {{p 3}^{2}}}+{{{p 1}^{2}}\ {{p 2}^{2}}}\right)}\  q 0}\right)}\ {{q 2}^{3}}}+{{\left({{\left({p 0 \ {{p 2}^{3}}}-{2 \  p 0 \ {{p 1}^{2}}\  p 2}\right)}\ {{q 1}^{2}}}+{{\left({2 \  p 1 \  p 2 \ {{p 3}^{2}}}-{p 1 \ {{p 2}^{3}}}+{{\left({2 \ {{p 1}^{3}}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 2}\right)}\  q 0 \  q 1}+{{\left(-{2 \  p 0 \  p 2 \ {{p 3}^{2}}}-{2 \  p 0 \ {{p 1}^{2}}\  p 2}\right)}\ {{q 0}^{2}}}\right)}\ {{q 2}^{2}}}+{{\left({{\left({2 \  p 0 \  p 1 \ {{p 2}^{2}}}-{p 0 \ {{p 1}^{3}}}\right)}\ {{q 1}^{3}}}+{{\left({{{p 1}^{2}}\ {{p 3}^{2}}}+{{\left(-{2 \ {{p 1}^{2}}}-{2 \ {{p 0}^{2}}}\right)}\ {{p 2}^{2}}}+{{p 1}^{4}}+{2 \ {{p 0}^{2}}\ {{p 1}^{2}}}\right)}\  q 0 \ {{q 1}^{2}}}+{{\left(-{2 \  p 0 \  p 1 \ {{p 3}^{2}}}+{2 \  p 0 \  p 1 \ {{p 2}^{2}}}-{2 \  p 0 \ {{p 1}^{3}}}-{{{p 0}^{3}}\  p 1}\right)}\ {{q 0}^{2}}\  q 1}+{{\left({{{p 0}^{2}}\ {{p 3}^{2}}}+{{{p 0}^{2}}\ {{p 1}^{2}}}\right)}\ {{q 0}^{3}}}\right)}\  q 2}+{p 0 \ {{p 1}^{2}}\  p 2 \ {{q 1}^{4}}}+{{\left(-{{p 1}^{3}}-{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 2 \  q 0 \ {{q 1}^{3}}}+{{\left({2 \  p 0 \ {{p 1}^{2}}}+{{p 0}^{3}}\right)}\  p 2 \ {{q 0}^{2}}\ {{q 1}^{2}}}-{{{p 0}^{2}}\  p 1 \  p 2 \ {{q 0}^{3}}\  q 1}}&{-{p 1 \  p 2 \ {{p 3}^{2}}\ {{q 3}^{4}}}+{{\left({{\left({p 1 \ {{p 3}^{3}}}-{2 \  p 1 \ {{p 2}^{2}}\  p 3}\right)}\  q 2}+{{\left({p 2 \ {{p 3}^{3}}}-{2 \ {{p 1}^{2}}\  p 2 \  p 3}\right)}\  q 1}+{2 \  p 0 \  p 1 \  p 2 \  p 3 \  q 0}\right)}\ {{q 3}^{3}}}+{{\left({{\left({2 \  p 1 \  p 2 \ {{p 3}^{2}}}-{p 1 \ {{p 2}^{3}}}\right)}\ {{q 2}^{2}}}+{{\left({{\left(-{{p 3}^{4}}+{{\left({2 \ {{p 2}^{2}}}+{2 \ {{p 1}^{2}}}+{{p 0}^{2}}\right)}\ {{p 3}^{2}}}-{2 \ {{p 1}^{2}}\ {{p 2}^{2}}}\right)}\  q 1}+{{\left(-{3 \  p 0 \  p 1 \ {{p 3}^{2}}}+{2 \  p 0 \  p 1 \ {{p 2}^{2}}}\right)}\  q 0}\right)}\  q 2}+{{\left({2 \  p 1 \  p 2 \ {{p 3}^{2}}}-{{{p 1}^{3}}\  p 2}\right)}\ {{q 1}^{2}}}+{{\left(-{3 \  p 0 \  p 2 \ {{p 3}^{2}}}+{2 \  p 0 \ {{p 1}^{2}}\  p 2}\right)}\  q 0 \  q 1}+{{\left({p 1 \  p 2 \ {{p 3}^{2}}}-{{{p 0}^{2}}\  p 1 \  p 2}\right)}\ {{q 0}^{2}}}\right)}\ {{q 3}^{2}}}+{{\left({p 1 \ {{p 2}^{2}}\  p 3 \ {{q 2}^{3}}}+{{\left({{\left(-{2 \  p 2 \ {{p 3}^{3}}}+{{\left({{p 2}^{3}}+{{\left({2 \ {{p 1}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\  p 2}\right)}\  p 3}\right)}\  q 1}-{4 \  p 0 \  p 1 \  p 2 \  p 3 \  q 0}\right)}\ {{q 2}^{2}}}+{{\left({{\left(-{2 \  p 1 \ {{p 3}^{3}}}+{{\left({2 \  p 1 \ {{p 2}^{2}}}+{{p 1}^{3}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 3}\right)}\ {{q 1}^{2}}}+{{\left({2 \  p 0 \ {{p 3}^{3}}}+{{\left(-{4 \  p 0 \ {{p 2}^{2}}}-{4 \  p 0 \ {{p 1}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 3}\right)}\  q 0 \  q 1}+{{\left({2 \  p 1 \ {{p 2}^{2}}}+{3 \ {{p 0}^{2}}\  p 1}\right)}\  p 3 \ {{q 0}^{2}}}\right)}\  q 2}+{{{p 1}^{2}}\  p 2 \  p 3 \ {{q 1}^{3}}}-{4 \  p 0 \  p 1 \  p 2 \  p 3 \  q 0 \ {{q 1}^{2}}}+{{\left({2 \ {{p 1}^{2}}}+{3 \ {{p 0}^{2}}}\right)}\  p 2 \  p 3 \ {{q 0}^{2}}\  q 1}-{2 \  p 0 \  p 1 \  p 2 \  p 3 \ {{q 0}^{3}}}\right)}\  q 3}+{{\left({{\left(-{{{p 2}^{2}}\ {{p 3}^{2}}}+{{{p 0}^{2}}\ {{p 2}^{2}}}\right)}\  q 1}-{p 0 \  p 1 \ {{p 2}^{2}}\  q 0}\right)}\ {{q 2}^{3}}}+{{\left({{\left(-{2 \  p 1 \  p 2 \ {{p 3}^{2}}}+{2 \ {{p 0}^{2}}\  p 1 \  p 2}\right)}\ {{q 1}^{2}}}+{{\left({2 \  p 0 \  p 2 \ {{p 3}^{2}}}-{p 0 \ {{p 2}^{3}}}+{{\left(-{2 \  p 0 \ {{p 1}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 2}\right)}\  q 0 \  q 1}+{{\left({p 1 \ {{p 2}^{3}}}+{2 \ {{p 0}^{2}}\  p 1 \  p 2}\right)}\ {{q 0}^{2}}}\right)}\ {{q 2}^{2}}}+{{\left({{\left(-{{{p 1}^{2}}\ {{p 3}^{2}}}+{{{p 0}^{2}}\ {{p 1}^{2}}}\right)}\ {{q 1}^{3}}}+{{\left({2 \  p 0 \  p 1 \ {{p 3}^{2}}}-{2 \  p 0 \  p 1 \ {{p 2}^{2}}}-{p 0 \ {{p 1}^{3}}}-{2 \ {{p 0}^{3}}\  p 1}\right)}\  q 0 \ {{q 1}^{2}}}+{{\left(-{{{p 0}^{2}}\ {{p 3}^{2}}}+{{\left({2 \ {{p 1}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\ {{p 2}^{2}}}+{2 \ {{p 0}^{2}}\ {{p 1}^{2}}}+{{p 0}^{4}}\right)}\ {{q 0}^{2}}\  q 1}+{{\left(-{2 \  p 0 \  p 1 \ {{p 2}^{2}}}-{{{p 0}^{3}}\  p 1}\right)}\ {{q 0}^{3}}}\right)}\  q 2}-{p 0 \ {{p 1}^{2}}\  p 2 \  q 0 \ {{q 1}^{3}}}+{{\left({{p 1}^{3}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 2 \ {{q 0}^{2}}\ {{q 1}^{2}}}+{{\left(-{2 \  p 0 \ {{p 1}^{2}}}-{{p 0}^{3}}\right)}\  p 2 \ {{q 0}^{3}}\  q 1}+{{{p 0}^{2}}\  p 1 \  p 2 \ {{q 0}^{4}}}}&{-{{{p 2}^{2}}\ {{p 3}^{2}}\ {{q 3}^{4}}}+{{\left({{\left({2 \  p 2 \ {{p 3}^{3}}}-{2 \ {{p 2}^{3}}\  p 3}\right)}\  q 2}-{2 \  p 1 \ {{p 2}^{2}}\  p 3 \  q 1}+{2 \  p 0 \ {{p 2}^{2}}\  p 3 \  q 0}\right)}\ {{q 3}^{3}}}+{{\left({{\left(-{{p 3}^{4}}+{{\left({4 \ {{p 2}^{2}}}-{{p 1}^{2}}+{{p 0}^{2}}\right)}\ {{p 3}^{2}}}-{{p 2}^{4}}\right)}\ {{q 2}^{2}}}+{{\left({{\left({6 \  p 1 \  p 2 \ {{p 3}^{2}}}-{2 \  p 1 \ {{p 2}^{3}}}\right)}\  q 1}+{{\left(-{6 \  p 0 \  p 2 \ {{p 3}^{2}}}+{2 \  p 0 \ {{p 2}^{3}}}\right)}\  q 0}\right)}\  q 2}+{{\left(-{{{p 2}^{2}}\ {{p 3}^{2}}}-{{{p 1}^{2}}\ {{p 2}^{2}}}\right)}\ {{q 1}^{2}}}+{2 \  p 0 \  p 1 \ {{p 2}^{2}}\  q 0 \  q 1}+{{\left({{{p 2}^{2}}\ {{p 3}^{2}}}-{{{p 0}^{2}}\ {{p 2}^{2}}}\right)}\ {{q 0}^{2}}}\right)}\ {{q 3}^{2}}}+{{\left({{\left(-{2 \  p 2 \ {{p 3}^{3}}}+{{\left({2 \ {{p 2}^{3}}}+{{\left(-{2 \ {{p 1}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\  p 2}\right)}\  p 3}\right)}\ {{q 2}^{3}}}+{{\left({{\left(-{2 \  p 1 \ {{p 3}^{3}}}+{{\left({8 \  p 1 \ {{p 2}^{2}}}-{2 \ {{p 1}^{3}}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 3}\right)}\  q 1}+{{\left({2 \  p 0 \ {{p 3}^{3}}}+{{\left(-{8 \  p 0 \ {{p 2}^{2}}}+{2 \  p 0 \ {{p 1}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 3}\right)}\  q 0}\right)}\ {{q 2}^{2}}}+{{\left({{\left(-{2 \ {{p 2}^{3}}}+{6 \ {{p 1}^{2}}\  p 2}\right)}\  p 3 \ {{q 1}^{2}}}-{{12}\  p 0 \  p 1 \  p 2 \  p 3 \  q 0 \  q 1}+{{\left({2 \ {{p 2}^{3}}}+{6 \ {{p 0}^{2}}\  p 2}\right)}\  p 3 \ {{q 0}^{2}}}\right)}\  q 2}-{2 \  p 1 \ {{p 2}^{2}}\  p 3 \ {{q 1}^{3}}}+{2 \  p 0 \ {{p 2}^{2}}\  p 3 \  q 0 \ {{q 1}^{2}}}+{2 \  p 1 \ {{p 2}^{2}}\  p 3 \ {{q 0}^{2}}\  q 1}-{2 \  p 0 \ {{p 2}^{2}}\  p 3 \ {{q 0}^{3}}}\right)}\  q 3}+{{\left(-{{{p 2}^{2}}\ {{p 3}^{2}}}+{{\left(-{{p 1}^{2}}+{{p 0}^{2}}\right)}\ {{p 2}^{2}}}\right)}\ {{q 2}^{4}}}+{{\left({{\left(-{2 \  p 1 \  p 2 \ {{p 3}^{2}}}+{2 \  p 1 \ {{p 2}^{3}}}+{{\left(-{2 \ {{p 1}^{3}}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 2}\right)}\  q 1}+{{\left({2 \  p 0 \  p 2 \ {{p 3}^{2}}}-{2 \  p 0 \ {{p 2}^{3}}}+{{\left({2 \  p 0 \ {{p 1}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 2}\right)}\  q 0}\right)}\ {{q 2}^{3}}}+{{\left({{\left(-{{{p 1}^{2}}\ {{p 3}^{2}}}-{{p 2}^{4}}+{4 \ {{p 1}^{2}}\ {{p 2}^{2}}}-{{p 1}^{4}}+{{{p 0}^{2}}\ {{p 1}^{2}}}\right)}\ {{q 1}^{2}}}+{{\left({2 \  p 0 \  p 1 \ {{p 3}^{2}}}-{8 \  p 0 \  p 1 \ {{p 2}^{2}}}+{2 \  p 0 \ {{p 1}^{3}}}-{2 \ {{p 0}^{3}}\  p 1}\right)}\  q 0 \  q 1}+{{\left(-{{{p 0}^{2}}\ {{p 3}^{2}}}+{{p 2}^{4}}+{4 \ {{p 0}^{2}}\ {{p 2}^{2}}}-{{{p 0}^{2}}\ {{p 1}^{2}}}+{{p 0}^{4}}\right)}\ {{q 0}^{2}}}\right)}\ {{q 2}^{2}}}+{{\left({{\left(-{2 \  p 1 \ {{p 2}^{3}}}+{2 \ {{p 1}^{3}}\  p 2}\right)}\ {{q 1}^{3}}}+{{\left({2 \  p 0 \ {{p 2}^{3}}}-{6 \  p 0 \ {{p 1}^{2}}\  p 2}\right)}\  q 0 \ {{q 1}^{2}}}+{{\left({2 \  p 1 \ {{p 2}^{3}}}+{6 \ {{p 0}^{2}}\  p 1 \  p 2}\right)}\ {{q 0}^{2}}\  q 1}+{{\left(-{2 \  p 0 \ {{p 2}^{3}}}-{2 \ {{p 0}^{3}}\  p 2}\right)}\ {{q 0}^{3}}}\right)}\  q 2}-{{{p 1}^{2}}\ {{p 2}^{2}}\ {{q 1}^{4}}}+{2 \  p 0 \  p 1 \ {{p 2}^{2}}\  q 0 \ {{q 1}^{3}}}+{{\left({{p 1}^{2}}-{{p 0}^{2}}\right)}\ {{p 2}^{2}}\ {{q 0}^{2}}\ {{q 1}^{2}}}-{2 \  p 0 \  p 1 \ {{p 2}^{2}}\ {{q 0}^{3}}\  q 1}+{{{p 0}^{2}}\ {{p 2}^{2}}\ {{q 0}^{4}}}}&{{{\left({{\left(-{{p 1}^{2}}+{{p 0}^{2}}\right)}\ {{p 3}^{2}}\  q 2}+{p 1 \  p 2 \ {{p 3}^{2}}\  q 1}-{p 0 \  p 2 \ {{p 3}^{2}}\  q 0}\right)}\ {{q 3}^{3}}}+{{\left({{\left(-{2 \ {{p 1}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\  p 2 \  p 3 \ {{q 2}^{2}}}+{{\left({{\left({p 1 \ {{p 3}^{3}}}+{{\left({2 \  p 1 \ {{p 2}^{2}}}-{2 \ {{p 1}^{3}}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 3}\right)}\  q 1}+{{\left(-{p 0 \ {{p 3}^{3}}}+{{\left(-{2 \  p 0 \ {{p 2}^{2}}}+{2 \  p 0 \ {{p 1}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 3}\right)}\  q 0}\right)}\  q 2}+{{\left(-{p 2 \ {{p 3}^{3}}}+{2 \ {{p 1}^{2}}\  p 2 \  p 3}\right)}\ {{q 1}^{2}}}-{4 \  p 0 \  p 1 \  p 2 \  p 3 \  q 0 \  q 1}+{{\left({p 2 \ {{p 3}^{3}}}+{2 \ {{p 0}^{2}}\  p 2 \  p 3}\right)}\ {{q 0}^{2}}}\right)}\ {{q 3}^{2}}}+{{\left({{\left(-{{p 1}^{2}}+{{p 0}^{2}}\right)}\ {{p 2}^{2}}\ {{q 2}^{3}}}+{{\left({{\left({2 \  p 1 \  p 2 \ {{p 3}^{2}}}+{p 1 \ {{p 2}^{3}}}+{{\left(-{2 \ {{p 1}^{3}}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 2}\right)}\  q 1}+{{\left(-{2 \  p 0 \  p 2 \ {{p 3}^{2}}}-{p 0 \ {{p 2}^{3}}}+{{\left({2 \  p 0 \ {{p 1}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 2}\right)}\  q 0}\right)}\ {{q 2}^{2}}}+{{\left({{\left({{\left(-{2 \ {{p 2}^{2}}}+{2 \ {{p 1}^{2}}}\right)}\ {{p 3}^{2}}}+{2 \ {{p 1}^{2}}\ {{p 2}^{2}}}-{{p 1}^{4}}+{{{p 0}^{2}}\ {{p 1}^{2}}}\right)}\ {{q 1}^{2}}}+{{\left(-{4 \  p 0 \  p 1 \ {{p 3}^{2}}}-{4 \  p 0 \  p 1 \ {{p 2}^{2}}}+{2 \  p 0 \ {{p 1}^{3}}}-{2 \ {{p 0}^{3}}\  p 1}\right)}\  q 0 \  q 1}+{{\left({{\left({2 \ {{p 2}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\ {{p 3}^{2}}}+{2 \ {{p 0}^{2}}\ {{p 2}^{2}}}-{{{p 0}^{2}}\ {{p 1}^{2}}}+{{p 0}^{4}}\right)}\ {{q 0}^{2}}}\right)}\  q 2}+{{\left(-{2 \  p 1 \  p 2 \ {{p 3}^{2}}}+{{{p 1}^{3}}\  p 2}\right)}\ {{q 1}^{3}}}+{{\left({2 \  p 0 \  p 2 \ {{p 3}^{2}}}-{3 \  p 0 \ {{p 1}^{2}}\  p 2}\right)}\  q 0 \ {{q 1}^{2}}}+{{\left({2 \  p 1 \  p 2 \ {{p 3}^{2}}}+{3 \ {{p 0}^{2}}\  p 1 \  p 2}\right)}\ {{q 0}^{2}}\  q 1}+{{\left(-{2 \  p 0 \  p 2 \ {{p 3}^{2}}}-{{{p 0}^{3}}\  p 2}\right)}\ {{q 0}^{3}}}\right)}\  q 3}+{{\left({p 1 \ {{p 2}^{2}}\  p 3 \  q 1}-{p 0 \ {{p 2}^{2}}\  p 3 \  q 0}\right)}\ {{q 2}^{3}}}+{{\left({{\left(-{{p 2}^{3}}+{2 \ {{p 1}^{2}}\  p 2}\right)}\  p 3 \ {{q 1}^{2}}}-{4 \  p 0 \  p 1 \  p 2 \  p 3 \  q 0 \  q 1}+{{\left({{p 2}^{3}}+{2 \ {{p 0}^{2}}\  p 2}\right)}\  p 3 \ {{q 0}^{2}}}\right)}\ {{q 2}^{2}}}+{{\left({{\left(-{2 \  p 1 \ {{p 2}^{2}}}+{{p 1}^{3}}\right)}\  p 3 \ {{q 1}^{3}}}+{{\left({2 \  p 0 \ {{p 2}^{2}}}-{3 \  p 0 \ {{p 1}^{2}}}\right)}\  p 3 \  q 0 \ {{q 1}^{2}}}+{{\left({2 \  p 1 \ {{p 2}^{2}}}+{3 \ {{p 0}^{2}}\  p 1}\right)}\  p 3 \ {{q 0}^{2}}\  q 1}+{{\left(-{2 \  p 0 \ {{p 2}^{2}}}-{{p 0}^{3}}\right)}\  p 3 \ {{q 0}^{3}}}\right)}\  q 2}-{{{p 1}^{2}}\  p 2 \  p 3 \ {{q 1}^{4}}}+{2 \  p 0 \  p 1 \  p 2 \  p 3 \  q 0 \ {{q 1}^{3}}}+{{\left({{p 1}^{2}}-{{p 0}^{2}}\right)}\  p 2 \  p 3 \ {{q 0}^{2}}\ {{q 1}^{2}}}-{2 \  p 0 \  p 1 \  p 2 \  p 3 \ {{q 0}^{3}}\  q 1}+{{{p 0}^{2}}\  p 2 \  p 3 \ {{q 0}^{4}}}}
\
{{{\left(-{p 0 \  p 2 \ {{p 3}^{2}}\  q 2}-{p 0 \  p 1 \ {{p 3}^{2}}\  q 1}+{{\left({{p 2}^{2}}+{{p 1}^{2}}\right)}\ {{p 3}^{2}}\  q 0}\right)}\ {{q 3}^{3}}}+{{\left({{\left({p 0 \ {{p 3}^{3}}}-{2 \  p 0 \ {{p 2}^{2}}\  p 3}\right)}\ {{q 2}^{2}}}+{{\left(-{4 \  p 0 \  p 1 \  p 2 \  p 3 \  q 1}+{{\left(-{p 2 \ {{p 3}^{3}}}+{{\left({2 \ {{p 2}^{3}}}+{{\left({2 \ {{p 1}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\  p 2}\right)}\  p 3}\right)}\  q 0}\right)}\  q 2}+{{\left({p 0 \ {{p 3}^{3}}}-{2 \  p 0 \ {{p 1}^{2}}\  p 3}\right)}\ {{q 1}^{2}}}+{{\left(-{p 1 \ {{p 3}^{3}}}+{{\left({2 \  p 1 \ {{p 2}^{2}}}+{2 \ {{p 1}^{3}}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 3}\right)}\  q 0 \  q 1}+{{\left(-{2 \  p 0 \ {{p 2}^{2}}}-{2 \  p 0 \ {{p 1}^{2}}}\right)}\  p 3 \ {{q 0}^{2}}}\right)}\ {{q 3}^{2}}}+{{\left({{\left({2 \  p 0 \  p 2 \ {{p 3}^{2}}}-{p 0 \ {{p 2}^{3}}}\right)}\ {{q 2}^{3}}}+{{\left({{\left({2 \  p 0 \  p 1 \ {{p 3}^{2}}}-{3 \  p 0 \  p 1 \ {{p 2}^{2}}}\right)}\  q 1}+{{\left({{\left(-{2 \ {{p 2}^{2}}}-{2 \ {{p 0}^{2}}}\right)}\ {{p 3}^{2}}}+{{p 2}^{4}}+{{\left({{p 1}^{2}}+{2 \ {{p 0}^{2}}}\right)}\ {{p 2}^{2}}}\right)}\  q 0}\right)}\ {{q 2}^{2}}}+{{\left({{\left({2 \  p 0 \  p 2 \ {{p 3}^{2}}}-{3 \  p 0 \ {{p 1}^{2}}\  p 2}\right)}\ {{q 1}^{2}}}+{{\left(-{4 \  p 1 \  p 2 \ {{p 3}^{2}}}+{2 \  p 1 \ {{p 2}^{3}}}+{{\left({2 \ {{p 1}^{3}}}+{4 \ {{p 0}^{2}}\  p 1}\right)}\  p 2}\right)}\  q 0 \  q 1}+{{\left({2 \  p 0 \  p 2 \ {{p 3}^{2}}}-{2 \  p 0 \ {{p 2}^{3}}}+{{\left(-{2 \  p 0 \ {{p 1}^{2}}}-{{p 0}^{3}}\right)}\  p 2}\right)}\ {{q 0}^{2}}}\right)}\  q 2}+{{\left({2 \  p 0 \  p 1 \ {{p 3}^{2}}}-{p 0 \ {{p 1}^{3}}}\right)}\ {{q 1}^{3}}}+{{\left({{\left(-{2 \ {{p 1}^{2}}}-{2 \ {{p 0}^{2}}}\right)}\ {{p 3}^{2}}}+{{{p 1}^{2}}\ {{p 2}^{2}}}+{{p 1}^{4}}+{2 \ {{p 0}^{2}}\ {{p 1}^{2}}}\right)}\  q 0 \ {{q 1}^{2}}}+{{\left({2 \  p 0 \  p 1 \ {{p 3}^{2}}}-{2 \  p 0 \  p 1 \ {{p 2}^{2}}}-{2 \  p 0 \ {{p 1}^{3}}}-{{{p 0}^{3}}\  p 1}\right)}\ {{q 0}^{2}}\  q 1}+{{\left({{{p 0}^{2}}\ {{p 2}^{2}}}+{{{p 0}^{2}}\ {{p 1}^{2}}}\right)}\ {{q 0}^{3}}}\right)}\  q 3}+{p 0 \ {{p 2}^{2}}\  p 3 \ {{q 2}^{4}}}+{{\left({2 \  p 0 \  p 1 \  p 2 \  p 3 \  q 1}+{{\left(-{{p 2}^{3}}-{2 \ {{p 0}^{2}}\  p 2}\right)}\  p 3 \  q 0}\right)}\ {{q 2}^{3}}}+{{\left({{\left({p 0 \ {{p 2}^{2}}}+{p 0 \ {{p 1}^{2}}}\right)}\  p 3 \ {{q 1}^{2}}}+{{\left(-{3 \  p 1 \ {{p 2}^{2}}}-{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 3 \  q 0 \  q 1}+{{\left({2 \  p 0 \ {{p 2}^{2}}}+{{p 0}^{3}}\right)}\  p 3 \ {{q 0}^{2}}}\right)}\ {{q 2}^{2}}}+{{\left({2 \  p 0 \  p 1 \  p 2 \  p 3 \ {{q 1}^{3}}}+{{\left(-{3 \ {{p 1}^{2}}}-{2 \ {{p 0}^{2}}}\right)}\  p 2 \  p 3 \  q 0 \ {{q 1}^{2}}}+{4 \  p 0 \  p 1 \  p 2 \  p 3 \ {{q 0}^{2}}\  q 1}-{{{p 0}^{2}}\  p 2 \  p 3 \ {{q 0}^{3}}}\right)}\  q 2}+{p 0 \ {{p 1}^{2}}\  p 3 \ {{q 1}^{4}}}+{{\left(-{{p 1}^{3}}-{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 3 \  q 0 \ {{q 1}^{3}}}+{{\left({2 \  p 0 \ {{p 1}^{2}}}+{{p 0}^{3}}\right)}\  p 3 \ {{q 0}^{2}}\ {{q 1}^{2}}}-{{{p 0}^{2}}\  p 1 \  p 3 \ {{q 0}^{3}}\  q 1}}&{{{\left({p 1 \  p 2 \ {{p 3}^{2}}\  q 2}+{{\left(-{{p 2}^{2}}+{{p 0}^{2}}\right)}\ {{p 3}^{2}}\  q 1}-{p 0 \  p 1 \ {{p 3}^{2}}\  q 0}\right)}\ {{q 3}^{3}}}+{{\left({{\left(-{p 1 \ {{p 3}^{3}}}+{2 \  p 1 \ {{p 2}^{2}}\  p 3}\right)}\ {{q 2}^{2}}}+{{\left({{\left({p 2 \ {{p 3}^{3}}}+{{\left(-{2 \ {{p 2}^{3}}}+{{\left({2 \ {{p 1}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\  p 2}\right)}\  p 3}\right)}\  q 1}-{4 \  p 0 \  p 1 \  p 2 \  p 3 \  q 0}\right)}\  q 2}+{{\left(-{2 \  p 1 \ {{p 2}^{2}}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 3 \ {{q 1}^{2}}}+{{\left(-{p 0 \ {{p 3}^{3}}}+{{\left({2 \  p 0 \ {{p 2}^{2}}}-{2 \  p 0 \ {{p 1}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 3}\right)}\  q 0 \  q 1}+{{\left({p 1 \ {{p 3}^{3}}}+{2 \ {{p 0}^{2}}\  p 1 \  p 3}\right)}\ {{q 0}^{2}}}\right)}\ {{q 3}^{2}}}+{{\left({{\left(-{2 \  p 1 \  p 2 \ {{p 3}^{2}}}+{p 1 \ {{p 2}^{3}}}\right)}\ {{q 2}^{3}}}+{{\left({{\left({{\left({2 \ {{p 2}^{2}}}-{2 \ {{p 1}^{2}}}\right)}\ {{p 3}^{2}}}-{{p 2}^{4}}+{{\left({2 \ {{p 1}^{2}}}+{{p 0}^{2}}\right)}\ {{p 2}^{2}}}\right)}\  q 1}+{{\left({2 \  p 0 \  p 1 \ {{p 3}^{2}}}-{3 \  p 0 \  p 1 \ {{p 2}^{2}}}\right)}\  q 0}\right)}\ {{q 2}^{2}}}+{{\left({{\left({2 \  p 1 \  p 2 \ {{p 3}^{2}}}-{2 \  p 1 \ {{p 2}^{3}}}+{{\left({{p 1}^{3}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 2}\right)}\ {{q 1}^{2}}}+{{\left(-{4 \  p 0 \  p 2 \ {{p 3}^{2}}}+{2 \  p 0 \ {{p 2}^{3}}}+{{\left(-{4 \  p 0 \ {{p 1}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 2}\right)}\  q 0 \  q 1}+{{\left({2 \  p 1 \  p 2 \ {{p 3}^{2}}}+{3 \ {{p 0}^{2}}\  p 1 \  p 2}\right)}\ {{q 0}^{2}}}\right)}\  q 2}+{{\left(-{{{p 1}^{2}}\ {{p 2}^{2}}}+{{{p 0}^{2}}\ {{p 1}^{2}}}\right)}\ {{q 1}^{3}}}+{{\left(-{2 \  p 0 \  p 1 \ {{p 3}^{2}}}+{2 \  p 0 \  p 1 \ {{p 2}^{2}}}-{p 0 \ {{p 1}^{3}}}-{2 \ {{p 0}^{3}}\  p 1}\right)}\  q 0 \ {{q 1}^{2}}}+{{\left({{\left({2 \ {{p 1}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\ {{p 3}^{2}}}-{{{p 0}^{2}}\ {{p 2}^{2}}}+{2 \ {{p 0}^{2}}\ {{p 1}^{2}}}+{{p 0}^{4}}\right)}\ {{q 0}^{2}}\  q 1}+{{\left(-{2 \  p 0 \  p 1 \ {{p 3}^{2}}}-{{{p 0}^{3}}\  p 1}\right)}\ {{q 0}^{3}}}\right)}\  q 3}-{p 1 \ {{p 2}^{2}}\  p 3 \ {{q 2}^{4}}}+{{\left({{\left({{p 2}^{3}}-{2 \ {{p 1}^{2}}\  p 2}\right)}\  p 3 \  q 1}+{2 \  p 0 \  p 1 \  p 2 \  p 3 \  q 0}\right)}\ {{q 2}^{3}}}+{{\left({{\left({2 \  p 1 \ {{p 2}^{2}}}-{{p 1}^{3}}\right)}\  p 3 \ {{q 1}^{2}}}+{{\left(-{3 \  p 0 \ {{p 2}^{2}}}+{2 \  p 0 \ {{p 1}^{2}}}\right)}\  p 3 \  q 0 \  q 1}+{{\left({p 1 \ {{p 2}^{2}}}-{{{p 0}^{2}}\  p 1}\right)}\  p 3 \ {{q 0}^{2}}}\right)}\ {{q 2}^{2}}}+{{\left({{{p 1}^{2}}\  p 2 \  p 3 \ {{q 1}^{3}}}-{4 \  p 0 \  p 1 \  p 2 \  p 3 \  q 0 \ {{q 1}^{2}}}+{{\left({2 \ {{p 1}^{2}}}+{3 \ {{p 0}^{2}}}\right)}\  p 2 \  p 3 \ {{q 0}^{2}}\  q 1}-{2 \  p 0 \  p 1 \  p 2 \  p 3 \ {{q 0}^{3}}}\right)}\  q 2}-{p 0 \ {{p 1}^{2}}\  p 3 \  q 0 \ {{q 1}^{3}}}+{{\left({{p 1}^{3}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 3 \ {{q 0}^{2}}\ {{q 1}^{2}}}+{{\left(-{2 \  p 0 \ {{p 1}^{2}}}-{{p 0}^{3}}\right)}\  p 3 \ {{q 0}^{3}}\  q 1}+{{{p 0}^{2}}\  p 1 \  p 3 \ {{q 0}^{4}}}}&{{{\left({{\left(-{{p 1}^{2}}+{{p 0}^{2}}\right)}\ {{p 3}^{2}}\  q 2}+{p 1 \  p 2 \ {{p 3}^{2}}\  q 1}-{p 0 \  p 2 \ {{p 3}^{2}}\  q 0}\right)}\ {{q 3}^{3}}}+{{\left({{\left(-{2 \ {{p 1}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\  p 2 \  p 3 \ {{q 2}^{2}}}+{{\left({{\left({p 1 \ {{p 3}^{3}}}+{{\left({2 \  p 1 \ {{p 2}^{2}}}-{2 \ {{p 1}^{3}}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 3}\right)}\  q 1}+{{\left(-{p 0 \ {{p 3}^{3}}}+{{\left(-{2 \  p 0 \ {{p 2}^{2}}}+{2 \  p 0 \ {{p 1}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 3}\right)}\  q 0}\right)}\  q 2}+{{\left(-{p 2 \ {{p 3}^{3}}}+{2 \ {{p 1}^{2}}\  p 2 \  p 3}\right)}\ {{q 1}^{2}}}-{4 \  p 0 \  p 1 \  p 2 \  p 3 \  q 0 \  q 1}+{{\left({p 2 \ {{p 3}^{3}}}+{2 \ {{p 0}^{2}}\  p 2 \  p 3}\right)}\ {{q 0}^{2}}}\right)}\ {{q 3}^{2}}}+{{\left({{\left(-{{p 1}^{2}}+{{p 0}^{2}}\right)}\ {{p 2}^{2}}\ {{q 2}^{3}}}+{{\left({{\left({2 \  p 1 \  p 2 \ {{p 3}^{2}}}+{p 1 \ {{p 2}^{3}}}+{{\left(-{2 \ {{p 1}^{3}}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 2}\right)}\  q 1}+{{\left(-{2 \  p 0 \  p 2 \ {{p 3}^{2}}}-{p 0 \ {{p 2}^{3}}}+{{\left({2 \  p 0 \ {{p 1}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 2}\right)}\  q 0}\right)}\ {{q 2}^{2}}}+{{\left({{\left({{\left(-{2 \ {{p 2}^{2}}}+{2 \ {{p 1}^{2}}}\right)}\ {{p 3}^{2}}}+{2 \ {{p 1}^{2}}\ {{p 2}^{2}}}-{{p 1}^{4}}+{{{p 0}^{2}}\ {{p 1}^{2}}}\right)}\ {{q 1}^{2}}}+{{\left(-{4 \  p 0 \  p 1 \ {{p 3}^{2}}}-{4 \  p 0 \  p 1 \ {{p 2}^{2}}}+{2 \  p 0 \ {{p 1}^{3}}}-{2 \ {{p 0}^{3}}\  p 1}\right)}\  q 0 \  q 1}+{{\left({{\left({2 \ {{p 2}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\ {{p 3}^{2}}}+{2 \ {{p 0}^{2}}\ {{p 2}^{2}}}-{{{p 0}^{2}}\ {{p 1}^{2}}}+{{p 0}^{4}}\right)}\ {{q 0}^{2}}}\right)}\  q 2}+{{\left(-{2 \  p 1 \  p 2 \ {{p 3}^{2}}}+{{{p 1}^{3}}\  p 2}\right)}\ {{q 1}^{3}}}+{{\left({2 \  p 0 \  p 2 \ {{p 3}^{2}}}-{3 \  p 0 \ {{p 1}^{2}}\  p 2}\right)}\  q 0 \ {{q 1}^{2}}}+{{\left({2 \  p 1 \  p 2 \ {{p 3}^{2}}}+{3 \ {{p 0}^{2}}\  p 1 \  p 2}\right)}\ {{q 0}^{2}}\  q 1}+{{\left(-{2 \  p 0 \  p 2 \ {{p 3}^{2}}}-{{{p 0}^{3}}\  p 2}\right)}\ {{q 0}^{3}}}\right)}\  q 3}+{{\left({p 1 \ {{p 2}^{2}}\  p 3 \  q 1}-{p 0 \ {{p 2}^{2}}\  p 3 \  q 0}\right)}\ {{q 2}^{3}}}+{{\left({{\left(-{{p 2}^{3}}+{2 \ {{p 1}^{2}}\  p 2}\right)}\  p 3 \ {{q 1}^{2}}}-{4 \  p 0 \  p 1 \  p 2 \  p 3 \  q 0 \  q 1}+{{\left({{p 2}^{3}}+{2 \ {{p 0}^{2}}\  p 2}\right)}\  p 3 \ {{q 0}^{2}}}\right)}\ {{q 2}^{2}}}+{{\left({{\left(-{2 \  p 1 \ {{p 2}^{2}}}+{{p 1}^{3}}\right)}\  p 3 \ {{q 1}^{3}}}+{{\left({2 \  p 0 \ {{p 2}^{2}}}-{3 \  p 0 \ {{p 1}^{2}}}\right)}\  p 3 \  q 0 \ {{q 1}^{2}}}+{{\left({2 \  p 1 \ {{p 2}^{2}}}+{3 \ {{p 0}^{2}}\  p 1}\right)}\  p 3 \ {{q 0}^{2}}\  q 1}+{{\left(-{2 \  p 0 \ {{p 2}^{2}}}-{{p 0}^{3}}\right)}\  p 3 \ {{q 0}^{3}}}\right)}\  q 2}-{{{p 1}^{2}}\  p 2 \  p 3 \ {{q 1}^{4}}}+{2 \  p 0 \  p 1 \  p 2 \  p 3 \  q 0 \ {{q 1}^{3}}}+{{\left({{p 1}^{2}}-{{p 0}^{2}}\right)}\  p 2 \  p 3 \ {{q 0}^{2}}\ {{q 1}^{2}}}-{2 \  p 0 \  p 1 \  p 2 \  p 3 \ {{q 0}^{3}}\  q 1}+{{{p 0}^{2}}\  p 2 \  p 3 \ {{q 0}^{4}}}}&{{{\left(-{{p 2}^{2}}-{{p 1}^{2}}+{{p 0}^{2}}\right)}\ {{p 3}^{2}}\ {{q 3}^{4}}}+{{\left({{\left({2 \  p 2 \ {{p 3}^{3}}}+{{\left(-{2 \ {{p 2}^{3}}}+{{\left(-{2 \ {{p 1}^{2}}}+{2 \ {{p 0}^{2}}}\right)}\  p 2}\right)}\  p 3}\right)}\  q 2}+{{\left({2 \  p 1 \ {{p 3}^{3}}}+{{\left(-{2 \  p 1 \ {{p 2}^{2}}}-{2 \ {{p 1}^{3}}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 3}\right)}\  q 1}+{{\left(-{2 \  p 0 \ {{p 3}^{3}}}+{{\left({2 \  p 0 \ {{p 2}^{2}}}+{2 \  p 0 \ {{p 1}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 3}\right)}\  q 0}\right)}\ {{q 3}^{3}}}+{{\left({{\left(-{{p 3}^{4}}+{4 \ {{p 2}^{2}}\ {{p 3}^{2}}}-{{p 2}^{4}}+{{\left(-{{p 1}^{2}}+{{p 0}^{2}}\right)}\ {{p 2}^{2}}}\right)}\ {{q 2}^{2}}}+{{\left({{\left({8 \  p 1 \  p 2 \ {{p 3}^{2}}}-{2 \  p 1 \ {{p 2}^{3}}}+{{\left(-{2 \ {{p 1}^{3}}}+{2 \ {{p 0}^{2}}\  p 1}\right)}\  p 2}\right)}\  q 1}+{{\left(-{8 \  p 0 \  p 2 \ {{p 3}^{2}}}+{2 \  p 0 \ {{p 2}^{3}}}+{{\left({2 \  p 0 \ {{p 1}^{2}}}-{2 \ {{p 0}^{3}}}\right)}\  p 2}\right)}\  q 0}\right)}\  q 2}+{{\left(-{{p 3}^{4}}+{4 \ {{p 1}^{2}}\ {{p 3}^{2}}}-{{{p 1}^{2}}\ {{p 2}^{2}}}-{{p 1}^{4}}+{{{p 0}^{2}}\ {{p 1}^{2}}}\right)}\ {{q 1}^{2}}}+{{\left(-{8 \  p 0 \  p 1 \ {{p 3}^{2}}}+{2 \  p 0 \  p 1 \ {{p 2}^{2}}}+{2 \  p 0 \ {{p 1}^{3}}}-{2 \ {{p 0}^{3}}\  p 1}\right)}\  q 0 \  q 1}+{{\left({{p 3}^{4}}+{4 \ {{p 0}^{2}}\ {{p 3}^{2}}}-{{{p 0}^{2}}\ {{p 2}^{2}}}-{{{p 0}^{2}}\ {{p 1}^{2}}}+{{p 0}^{4}}\right)}\ {{q 0}^{2}}}\right)}\ {{q 3}^{2}}}+{{\left({{\left(-{2 \  p 2 \ {{p 3}^{3}}}+{2 \ {{p 2}^{3}}\  p 3}\right)}\ {{q 2}^{3}}}+{{\left({{\left(-{2 \  p 1 \ {{p 3}^{3}}}+{6 \  p 1 \ {{p 2}^{2}}\  p 3}\right)}\  q 1}+{{\left({2 \  p 0 \ {{p 3}^{3}}}-{6 \  p 0 \ {{p 2}^{2}}\  p 3}\right)}\  q 0}\right)}\ {{q 2}^{2}}}+{{\left({{\left(-{2 \  p 2 \ {{p 3}^{3}}}+{6 \ {{p 1}^{2}}\  p 2 \  p 3}\right)}\ {{q 1}^{2}}}-{{12}\  p 0 \  p 1 \  p 2 \  p 3 \  q 0 \  q 1}+{{\left({2 \  p 2 \ {{p 3}^{3}}}+{6 \ {{p 0}^{2}}\  p 2 \  p 3}\right)}\ {{q 0}^{2}}}\right)}\  q 2}+{{\left(-{2 \  p 1 \ {{p 3}^{3}}}+{2 \ {{p 1}^{3}}\  p 3}\right)}\ {{q 1}^{3}}}+{{\left({2 \  p 0 \ {{p 3}^{3}}}-{6 \  p 0 \ {{p 1}^{2}}\  p 3}\right)}\  q 0 \ {{q 1}^{2}}}+{{\left({2 \  p 1 \ {{p 3}^{3}}}+{6 \ {{p 0}^{2}}\  p 1 \  p 3}\right)}\ {{q 0}^{2}}\  q 1}+{{\left(-{2 \  p 0 \ {{p 3}^{3}}}-{2 \ {{p 0}^{3}}\  p 3}\right)}\ {{q 0}^{3}}}\right)}\  q 3}-{{{p 2}^{2}}\ {{p 3}^{2}}\ {{q 2}^{4}}}+{{\left(-{2 \  p 1 \  p 2 \ {{p 3}^{2}}\  q 1}+{2 \  p 0 \  p 2 \ {{p 3}^{2}}\  q 0}\right)}\ {{q 2}^{3}}}+{{\left({{\left(-{{p 2}^{2}}-{{p 1}^{2}}\right)}\ {{p 3}^{2}}\ {{q 1}^{2}}}+{2 \  p 0 \  p 1 \ {{p 3}^{2}}\  q 0 \  q 1}+{{\left({{p 2}^{2}}-{{p 0}^{2}}\right)}\ {{p 3}^{2}}\ {{q 0}^{2}}}\right)}\ {{q 2}^{2}}}+{{\left(-{2 \  p 1 \  p 2 \ {{p 3}^{2}}\ {{q 1}^{3}}}+{2 \  p 0 \  p 2 \ {{p 3}^{2}}\  q 0 \ {{q 1}^{2}}}+{2 \  p 1 \  p 2 \ {{p 3}^{2}}\ {{q 0}^{2}}\  q 1}-{2 \  p 0 \  p 2 \ {{p 3}^{2}}\ {{q 0}^{3}}}\right)}\  q 2}-{{{p 1}^{2}}\ {{p 3}^{2}}\ {{q 1}^{4}}}+{2 \  p 0 \  p 1 \ {{p 3}^{2}}\  q 0 \ {{q 1}^{3}}}+{{\left({{p 1}^{2}}-{{p 0}^{2}}\right)}\ {{p 3}^{2}}\ {{q 0}^{2}}\ {{q 1}^{2}}}-{2 \  p 0 \  p 1 \ {{p 3}^{2}}\ {{q 0}^{3}}\  q 1}+{{{p 0}^{2}}\ {{p 3}^{2}}\ {{q 0}^{4}}}}
(51)
Type: Matrix(Expression(Integer))
fricas
trace((PP*QQ-QQ*PP))

\label{eq52}0(52)
Type: Expression(Integer)

Unit

fricas
-(PP*QQ+QQ*PP)+PP+QQ;
Type: Matrix(Expression(Integer))
fricas
2/trace(%)*%;
Type: Matrix(Expression(Integer))
fricas
trace %

\label{eq53}2(53)
Type: Expression(Integer)
fricas
n:=map(Simplify,-(SS*TT+TT*SS)+SS+TT);
Type: Matrix(Expression(Integer))
fricas
Is?(n = (SS-TT)*(SS-TT) )

\label{eq54} \mbox{\rm true} (54)
Type: Boolean
fricas
N:=map(Simplify,2/trace(n)*n);
Type: Matrix(Expression(Integer))
fricas
Simplify trace N

\label{eq55}2(55)
Type: Expression(Integer)
fricas
Is?(N*N=N)

\label{eq56} \mbox{\rm true} (56)
Type: Boolean
fricas
Is?(SS*N=SS)

\label{eq57} \mbox{\rm true} (57)
Type: Boolean
fricas
Is?(TT*N=TT)

\label{eq58} \mbox{\rm true} (58)
Type: Boolean
fricas
Is?(N*SS=SS)

\label{eq59} \mbox{\rm true} (59)
Type: Boolean
fricas
Is?(N*TT=TT)

\label{eq60} \mbox{\rm true} (60)
Type: Boolean

fricas
-(PP*RR+RR*PP)+PP+RR;
Type: Matrix(Expression(Integer))
fricas
2/trace(%)*%

\label{eq61}\left[ 
\begin{array}{cccc}
{{{2 \ {{p 0}^{2}}}- 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}& 0 & 0 & 0 
\
0 &{{2 \ {{p 1}^{2}}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \  p 1 \  p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \  p 1 \  p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}
\
0 &{{2 \  p 1 \  p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \ {{p 2}^{2}}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \  p 2 \  p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}
\
0 &{{2 \  p 1 \  p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \  p 2 \  p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \ {{p 3}^{2}}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}
(61)
Type: Matrix(Expression(Integer))
fricas
-(SS*RR+RR*SS)+SS+RR;
Type: Matrix(Expression(Integer))
fricas
2/trace(%)*%

\label{eq62}\left[ 
\begin{array}{cccc}
1 & 0 & 0 & 0 
\
0 &{{{s 1}^{2}}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{s 1 \  s 2}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{s 1 \  s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}
\
0 &{{s 1 \  s 2}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{{s 2}^{2}}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{s 2 \  s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}
\
0 &{{s 1 \  s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{s 2 \  s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{{s 3}^{2}}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}
(62)
Type: Matrix(Expression(Integer))
fricas
-(UU*RR+RR*UU)+UU+RR;
Type: Matrix(Expression(Integer))
fricas
map(Simplify,2/trace(%)*%)

\label{eq63}\left[ 
\begin{array}{cccc}
1 & 0 & 0 & 0 
\
0 & 1 & 0 & 0 
\
0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
(63)
Type: Matrix(Expression(Integer))
fricas
-(UU*VV+VV*UU)+UU+VV;
Type: Matrix(Expression(Integer))
fricas
map(Simplify,2/trace(%)*%)

\label{eq64}\left[ 
\begin{array}{cccc}
1 & 0 & 0 & 0 
\
0 & 1 & 0 & 0 
\
0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
(64)
Type: Matrix(Expression(Integer))

Momentum

fricas
m:=map(x+->factor(numer x)/factor(denom x),-(PP*QQ+QQ*PP)/dot(P,Q)+PP+QQ)

\label{eq65}\left[ 
\begin{array}{cccc}
{{\left(q 0 + p 0 \right)}^{2}}& -{{\left(q 0 + p 0 \right)}\ {\left(q 1 + p 1 \right)}}& -{{\left(q 0 + p 0 \right)}\ {\left(q 2 + p 2 \right)}}& -{{\left(q 0 + p 0 \right)}\ {\left(q 3 + p 3 \right)}}
\
{{\left(q 0 + p 0 \right)}\ {\left(q 1 + p 1 \right)}}& -{{\left(q 1 + p 1 \right)}^{2}}& -{{\left(q 1 + p 1 \right)}\ {\left(q 2 + p 2 \right)}}& -{{\left(q 1 + p 1 \right)}\ {\left(q 3 + p 3 \right)}}
\
{{\left(q 0 + p 0 \right)}\ {\left(q 2 + p 2 \right)}}& -{{\left(q 1 + p 1 \right)}\ {\left(q 2 + p 2 \right)}}& -{{\left(q 2 + p 2 \right)}^{2}}& -{{\left(q 2 + p 2 \right)}\ {\left(q 3 + p 3 \right)}}
\
{{\left(q 0 + p 0 \right)}\ {\left(q 3 + p 3 \right)}}& -{{\left(q 1 + p 1 \right)}\ {\left(q 3 + p 3 \right)}}& -{{\left(q 2 + p 2 \right)}\ {\left(q 3 + p 3 \right)}}& -{{\left(q 3 + p 3 \right)}^{2}}
(65)
Type: Matrix(Fraction(Factored(SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer))))))
fricas
Is?(m = tensor(-(P+Q),(P+Q)))

\label{eq66} \mbox{\rm true} (66)
Type: Boolean
fricas
factor trace m

\label{eq67}\begin{array}{@{}l}
\displaystyle
-{\left({
\begin{array}{@{}l}
\displaystyle
{{q 3}^{2}}+{2 \  p 3 \  q 3}+{{q 2}^{2}}+{2 \  p 2 \  q 2}+{{q 1}^{2}}+{2 \  p 1 \  q 1}- 
\
\
\displaystyle
{{q 0}^{2}}-{2 \  p 0 \  q 0}+{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}
(67)
Type: Factored(SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer))))
fricas
M:=1/trace(m)*m::Matrix Scalar;
Type: Matrix(Expression(Integer))
fricas
trace M

\label{eq68}1(68)
Type: Expression(Integer)
fricas
Is?(M*M=M)

\label{eq69} \mbox{\rm true} (69)
Type: Boolean

fricas
(PP*RR+RR*PP)/dot(P,R)-PP-RR;
Type: Matrix(Expression(Integer))
fricas
1/trace(%)*%

\label{eq70}\left[ 
\begin{array}{cccc}
{{-{{p 0}^{2}}-{2 \  p 0}- 1}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \  p 0}- 1}}&{{{\left(p 0 + 1 \right)}\  p 1}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \  p 0}- 1}}&{{{\left(p 0 + 1 \right)}\  p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \  p 0}- 1}}&{{{\left(p 0 + 1 \right)}\  p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \  p 0}- 1}}
\
{{{\left(- p 0 - 1 \right)}\  p 1}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \  p 0}- 1}}&{{{p 1}^{2}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \  p 0}- 1}}&{{p 1 \  p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \  p 0}- 1}}&{{p 1 \  p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \  p 0}- 1}}
\
{{{\left(- p 0 - 1 \right)}\  p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \  p 0}- 1}}&{{p 1 \  p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \  p 0}- 1}}&{{{p 2}^{2}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \  p 0}- 1}}&{{p 2 \  p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \  p 0}- 1}}
\
{{{\left(- p 0 - 1 \right)}\  p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \  p 0}- 1}}&{{p 1 \  p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \  p 0}- 1}}&{{p 2 \  p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \  p 0}- 1}}&{{{p 3}^{2}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \  p 0}- 1}}
(70)
Type: Matrix(Expression(Integer))
fricas
trace %

\label{eq71}1(71)
Type: Expression(Integer)
fricas
--(SS*RR+RR*SS)/dot(S,R)-SS-RR;
--map(Simplify,1/trace(%)*%)
--(UU*VV+VV*UU)/dot(U,V)-UU-VV;
--1/trace(%)*%
(UU*RR+RR*UU)/dot(U,R)-UU-RR;
Type: Matrix(Expression(Integer))
fricas
map(Simplify,1/trace(%)*%)

\label{eq72}\left[ 
\begin{array}{cccc}
{{{\cosh \left({3 \  u}\right)}+{4 \ {\cosh \left({2 \  u}\right)}}+{7 \ {\cosh \left({u}\right)}}+ 4}\over{{4 \ {\cosh \left({2 \  u}\right)}}+{8 \ {\cosh \left({u}\right)}}+ 4}}&{{-{\sinh \left({3 \  u}\right)}-{2 \ {\sinh \left({2 \  u}\right)}}-{\sinh \left({u}\right)}}\over{{4 \ {\cosh \left({2 \  u}\right)}}+{8 \ {\cosh \left({u}\right)}}+ 4}}& 0 & 0 
\
{{{\sinh \left({3 \  u}\right)}+{2 \ {\sinh \left({2 \  u}\right)}}+{\sinh \left({u}\right)}}\over{{4 \ {\cosh \left({2 \  u}\right)}}+{8 \ {\cosh \left({u}\right)}}+ 4}}&{{-{\cosh \left({3 \  u}\right)}+{\cosh \left({u}\right)}}\over{{4 \ {\cosh \left({2 \  u}\right)}}+{8 \ {\cosh \left({u}\right)}}+ 4}}& 0 & 0 
\
0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
(72)
Type: Matrix(Expression(Integer))

Unit for 3 Observers

fricas
P:=1/sqrt(1-p1^2-p2^2-p3^2)*vect [1,-p1,-p2,-p3]

\label{eq73}\left[ 
\begin{array}{c}
{1 \over{\sqrt{-{{p 3}^{2}}-{{p 2}^{2}}-{{p 1}^{2}}+ 1}}}
\
-{p 1 \over{\sqrt{-{{p 3}^{2}}-{{p 2}^{2}}-{{p 1}^{2}}+ 1}}}
\
-{p 2 \over{\sqrt{-{{p 3}^{2}}-{{p 2}^{2}}-{{p 1}^{2}}+ 1}}}
\
-{p 3 \over{\sqrt{-{{p 3}^{2}}-{{p 2}^{2}}-{{p 1}^{2}}+ 1}}}
(73)
Type: Matrix(Expression(Integer))
fricas
PP:=tensor(-P,P)

\label{eq74}\left[ 
\begin{array}{cccc}
-{1 \over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}- 1}}& -{p 1 \over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}- 1}}& -{p 2 \over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}- 1}}& -{p 3 \over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}- 1}}
\
{p 1 \over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}- 1}}&{{{p 1}^{2}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}- 1}}&{{p 1 \  p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}- 1}}&{{p 1 \  p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}- 1}}
\
{p 2 \over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}- 1}}&{{p 1 \  p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}- 1}}&{{{p 2}^{2}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}- 1}}&{{p 2 \  p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}- 1}}
\
{p 3 \over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}- 1}}&{{p 1 \  p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}- 1}}&{{p 2 \  p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}- 1}}&{{{p 3}^{2}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}- 1}}
(74)
Type: Matrix(Expression(Integer))
fricas
Q:=1/sqrt(1-q1^2-q2^2-q3^2)*vect [1,-q1,-q2,-q3]

\label{eq75}\left[ 
\begin{array}{c}
{1 \over{\sqrt{-{{q 3}^{2}}-{{q 2}^{2}}-{{q 1}^{2}}+ 1}}}
\
-{q 1 \over{\sqrt{-{{q 3}^{2}}-{{q 2}^{2}}-{{q 1}^{2}}+ 1}}}
\
-{q 2 \over{\sqrt{-{{q 3}^{2}}-{{q 2}^{2}}-{{q 1}^{2}}+ 1}}}
\
-{q 3 \over{\sqrt{-{{q 3}^{2}}-{{q 2}^{2}}-{{q 1}^{2}}+ 1}}}
(75)
Type: Matrix(Expression(Integer))
fricas
QQ:=tensor(-Q,Q)

\label{eq76}\left[ 
\begin{array}{cccc}
-{1 \over{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}- 1}}& -{q 1 \over{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}- 1}}& -{q 2 \over{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}- 1}}& -{q 3 \over{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}- 1}}
\
{q 1 \over{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}- 1}}&{{{q 1}^{2}}\over{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}- 1}}&{{q 1 \  q 2}\over{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}- 1}}&{{q 1 \  q 3}\over{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}- 1}}
\
{q 2 \over{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}- 1}}&{{q 1 \  q 2}\over{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}- 1}}&{{{q 2}^{2}}\over{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}- 1}}&{{q 2 \  q 3}\over{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}- 1}}
\
{q 3 \over{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}- 1}}&{{q 1 \  q 3}\over{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}- 1}}&{{q 2 \  q 3}\over{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}- 1}}&{{{q 3}^{2}}\over{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}- 1}}
(76)
Type: Matrix(Expression(Integer))
fricas
R:=1/sqrt(1-r1^2-r2^2-r3^2)*vect [1,-r1,-r2,-r3]

\label{eq77}\left[ 
\begin{array}{c}
{1 \over{\sqrt{-{{r 3}^{2}}-{{r 2}^{2}}-{{r 1}^{2}}+ 1}}}
\
-{r 1 \over{\sqrt{-{{r 3}^{2}}-{{r 2}^{2}}-{{r 1}^{2}}+ 1}}}
\
-{r 2 \over{\sqrt{-{{r 3}^{2}}-{{r 2}^{2}}-{{r 1}^{2}}+ 1}}}
\
-{r 3 \over{\sqrt{-{{r 3}^{2}}-{{r 2}^{2}}-{{r 1}^{2}}+ 1}}}
(77)
Type: Matrix(Expression(Integer))
fricas
RR:=tensor(-R,R)

\label{eq78}\left[ 
\begin{array}{cccc}
-{1 \over{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}- 1}}& -{r 1 \over{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}- 1}}& -{r 2 \over{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}- 1}}& -{r 3 \over{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}- 1}}
\
{r 1 \over{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}- 1}}&{{{r 1}^{2}}\over{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}- 1}}&{{r 1 \  r 2}\over{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}- 1}}&{{r 1 \  r 3}\over{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}- 1}}
\
{r 2 \over{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}- 1}}&{{r 1 \  r 2}\over{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}- 1}}&{{{r 2}^{2}}\over{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}- 1}}&{{r 2 \  r 3}\over{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}- 1}}
\
{r 3 \over{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}- 1}}&{{r 1 \  r 3}\over{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}- 1}}&{{r 2 \  r 3}\over{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}- 1}}&{{{r 3}^{2}}\over{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}- 1}}
(78)
Type: Matrix(Expression(Integer))
fricas
gamma(p,q) == - dot(p,q)
Type: Void
fricas
-- Silberstein
f0 := - determinant(matrix [[1,gamma(P,Q),gamma(P,R)], _
                       [gamma(P,Q),1,gamma(Q,R)], _
                       [gamma(P,R),gamma(Q,R),1]] )
fricas
Compiling function gamma with type (Matrix(Expression(Integer)),
      Matrix(Expression(Integer))) -> Expression(Integer)

\label{eq79}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{\left(-{{p 1}^{2}}+ 1 \right)}\ {{q 2}^{2}}}+{{\left({2 \  p 1 \  p 2 \  q 1}-{2 \  p 2}\right)}\  q 2}+ 
\
\
\displaystyle
{{\left(-{{p 2}^{2}}+ 1 \right)}\ {{q 1}^{2}}}-{2 \  p 1 \  q 1}+{{p 2}^{2}}+{{p 1}^{2}}
(79)
Type: Expression(Integer)
fricas
f:=gamma(P,Q)^2 + gamma(P,R)^2 + gamma(Q,R)^2 - 2*gamma(P,Q)*gamma(P,R)*gamma(Q,R) - 1

\label{eq80}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{\left(-{{p 1}^{2}}+ 1 \right)}\ {{q 2}^{2}}}+{{\left({2 \  p 1 \  p 2 \  q 1}-{2 \  p 2}\right)}\  q 2}+ 
\
\
\displaystyle
{{\left(-{{p 2}^{2}}+ 1 \right)}\ {{q 1}^{2}}}-{2 \  p 1 \  q 1}+{{p 2}^{2}}+{{p 1}^{2}}
(80)
Type: Expression(Integer)
fricas
test(f0 = f)

\label{eq81} \mbox{\rm true} (81)
Type: Boolean
fricas
--
nf:=map(Simplify, (-gamma(P,Q)*gamma(P,R)+gamma(Q,R))/gamma(Q,R)*RR*QQ + _
                 (-gamma(Q,R)*gamma(Q,P)+gamma(R,P))/gamma(R,P)*RR*PP + _
                 (-gamma(P,Q)*gamma(P,R)+gamma(Q,R))/gamma(Q,R)*QQ*RR + _
                 (-gamma(R,Q)*gamma(R,P)+gamma(Q,P))/gamma(Q,P)*QQ*PP + _
                 (-gamma(Q,R)*gamma(Q,P)+gamma(R,P))/gamma(R,P)*PP*RR + _
                 (-gamma(R,Q)*gamma(R,P)+gamma(Q,P))/gamma(Q,P)*PP*QQ + _
                 (gamma(Q,R)^2-1)*PP + _
                 (gamma(P,Q)^2-1)*RR + _
                 (gamma(P,R)^2-1)*QQ );
Type: Matrix(Expression(Integer))
fricas
N := 1/f*nf;
Type: Matrix(Expression(Integer))
fricas
--N:=map(Simplify,3/trace(n)*n);
Simplify trace N

\label{eq82}3(82)
Type: Expression(Integer)
fricas
Is?(N*N=N)

\label{eq83} \mbox{\rm true} (83)
Type: Boolean
fricas
map(possible,N*N-N)

\label{eq84}\left[ 
\begin{array}{cccc}
0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 
(84)
Type: Matrix(Expression(Integer))
fricas
Is?(PP*N=PP)

\label{eq85} \mbox{\rm true} (85)
Type: Boolean
fricas
Is?(QQ*N=QQ)

\label{eq86} \mbox{\rm true} (86)
Type: Boolean
fricas
Is?(RR*N=RR)

\label{eq87} \mbox{\rm true} (87)
Type: Boolean
fricas
Is?(N*PP=PP)

\label{eq88} \mbox{\rm true} (88)
Type: Boolean
fricas
Is?(N*QQ=QQ)

\label{eq89} \mbox{\rm true} (89)
Type: Boolean
fricas
Is?(N*RR=RR)

\label{eq90} \mbox{\rm true} (90)
Type: Boolean

Orthogonal Observers

fricas
PP' := N - (1/trace(PP))*PP

\label{eq91}(91)
Type: Matrix(Expression(Integer))
fricas
Is?(PP'*PP' = PP')

\label{eq92} \mbox{\rm true} (92)
Type: Boolean
fricas
trace PP'

\label{eq93}2(93)
Type: Expression(Integer)
fricas
Is?(PP'*PP = 0*PP)

\label{eq94} \mbox{\rm true} (94)
Type: Boolean
fricas
-- derivation
Is?(PP'*(QQ*RR) = (PP'*QQ)*RR + QQ*(PP'*RR))

\label{eq95} \mbox{\rm false} (95)
Type: Boolean




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