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The Pauli Algebra Cl(3) Is Frobenius In Many Ways

Linear operators over a 8-dimensional vector space representing Pauli algebra

Ref:

We need the Axiom LinearOperator library.

fricas
)library CARTEN ARITY CMONAL CPROP CLOP CALEY
CartesianTensor is now explicitly exposed in frame initial CartesianTensor will be automatically loaded when needed from /var/aw/var/LatexWiki/CARTEN.NRLIB/CARTEN Arity is now explicitly exposed in frame initial Arity will be automatically loaded when needed from /var/aw/var/LatexWiki/ARITY.NRLIB/ARITY ClosedMonoidal is now explicitly exposed in frame initial ClosedMonoidal will be automatically loaded when needed from /var/aw/var/LatexWiki/CMONAL.NRLIB/CMONAL ClosedProp is now explicitly exposed in frame initial ClosedProp will be automatically loaded when needed from /var/aw/var/LatexWiki/CPROP.NRLIB/CPROP ClosedLinearOperator is now explicitly exposed in frame initial ClosedLinearOperator will be automatically loaded when needed from /var/aw/var/LatexWiki/CLOP.NRLIB/CLOP CaleyDickson is now explicitly exposed in frame initial CaleyDickson will be automatically loaded when needed from /var/aw/var/LatexWiki/CALEY.NRLIB/CALEY

Use the following macros for convenient notation

fricas
-- summation
macro Σ(x,i,n)==reduce(+,[x for i in n])
Type: Void
fricas
-- list
macro Ξ(f,i,n)==[f for i in n]
Type: Void
fricas
-- subscript and superscripts
macro sb == subscript
Type: Void
fricas
macro sp == superscript
Type: Void

𝐋 is the domain of 8-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients.

fricas
dim:=8

\label{eq1}8(1)
Type: PositiveInteger?
fricas
macro ℒ == List
Type: Void
fricas
macro ℂ == CaleyDickson
Type: Void
fricas
macro ℚ == Expression Integer
Type: Void
fricas
𝐋 := ClosedLinearOperator(OVAR ['1,'i,'j,'k,'ij,'ik,'jk,'ijk], ℚ)

\label{eq2}\hbox{\axiomType{ClosedLinearOperator}\ } (\hbox{\axiomType{OrderedVariableList}\ } ([ 1, i , j , k , ij , ik , jk , ijk ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))(2)
Type: Type
fricas
𝐞:ℒ 𝐋      := basisOut()

\label{eq3}\left[{|_{\  1}}, \:{|_{\  i}}, \:{|_{\  j}}, \:{|_{\  k}}, \:{|_{\  ij}}, \:{|_{\  ik}}, \:{|_{\  jk}}, \:{|_{\  ijk}}\right](3)
Type: List(ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)))
fricas
𝐝:ℒ 𝐋      := basisIn()

\label{eq4}\left[{|^{\  1}}, \:{|^{\  i}}, \:{|^{\  j}}, \:{|^{\  k}}, \:{|^{\  ij}}, \:{|^{\  ik}}, \:{|^{\  jk}}, \:{|^{\  ijk}}\right](4)
Type: List(ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)))
fricas
I:𝐋:=[1]   -- identity for composition

\label{eq5}{|_{\  1}^{\  1}}+{|_{\  i}^{\  i}}+{|_{\  j}^{\  j}}+{|_{\  k}^{\  k}}+{|_{\  ij}^{\  ij}}+{|_{\  ik}^{\  ik}}+{|_{\  jk}^{\  jk}}+{|_{\  ijk}^{\  ijk}}(5)
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
X:𝐋:=[2,1] -- twist

\label{eq6}\begin{array}{@{}l}
\displaystyle
{|_{\  1 \  1}^{\  1 \  1}}+{|_{\  i \  1}^{\  1 \  i}}+{|_{\  j \  1}^{\  1 \  j}}+{|_{\  k \  1}^{\  1 \  k}}+{|_{\  ij \  1}^{\  1 \  ij}}+ 
\
\
\displaystyle
{|_{\  ik \  1}^{\  1 \  ik}}+{|_{\  jk \  1}^{\  1 \  jk}}+{|_{\  ijk \  1}^{\  1 \  ijk}}+{|_{\  1 \  i}^{\  i \  1}}+{|_{\  i \  i}^{\  i \  i}}+ 
\
\
\displaystyle
{|_{\  j \  i}^{\  i \  j}}+{|_{\  k \  i}^{\  i \  k}}+{|_{\  ij \  i}^{\  i \  ij}}+{|_{\  ik \  i}^{\  i \  ik}}+{|_{\  jk \  i}^{\  i \  jk}}+ 
\
\
\displaystyle
{|_{\  ijk \  i}^{\  i \  ijk}}+{|_{\  1 \  j}^{\  j \  1}}+{|_{\  i \  j}^{\  j \  i}}+{|_{\  j \  j}^{\  j \  j}}+{|_{\  k \  j}^{\  j \  k}}+ 
\
\
\displaystyle
{|_{\  ij \  j}^{\  j \  ij}}+{|_{\  ik \  j}^{\  j \  ik}}+{|_{\  jk \  j}^{\  j \  jk}}+{|_{\  ijk \  j}^{\  j \  ijk}}+ 
\
\
\displaystyle
{|_{\  1 \  k}^{\  k \  1}}+{|_{\  i \  k}^{\  k \  i}}+{|_{\  j \  k}^{\  k \  j}}+{|_{\  k \  k}^{\  k \  k}}+{|_{\  ij \  k}^{\  k \  ij}}+ 
\
\
\displaystyle
{|_{\  ik \  k}^{\  k \  ik}}+{|_{\  jk \  k}^{\  k \  jk}}+{|_{\  ijk \  k}^{\  k \  ijk}}+{|_{\  1 \  ij}^{\  ij \  1}}+ 
\
\
\displaystyle
{|_{\  i \  ij}^{\  ij \  i}}+{|_{\  j \  ij}^{\  ij \  j}}+{|_{\  k \  ij}^{\  ij \  k}}+{|_{\  ij \  ij}^{\  ij \  ij}}+ 
\
\
\displaystyle
{|_{\  ik \  ij}^{\  ij \  ik}}+{|_{\  jk \  ij}^{\  ij \  jk}}+{|_{\  ijk \  ij}^{\  ij \  ijk}}+{|_{\  1 \  ik}^{\  ik \  1}}+ 
\
\
\displaystyle
{|_{\  i \  ik}^{\  ik \  i}}+{|_{\  j \  ik}^{\  ik \  j}}+{|_{\  k \  ik}^{\  ik \  k}}+{|_{\  ij \  ik}^{\  ik \  ij}}+ 
\
\
\displaystyle
{|_{\  ik \  ik}^{\  ik \  ik}}+{|_{\  jk \  ik}^{\  ik \  jk}}+{|_{\  ijk \  ik}^{\  ik \  ijk}}+{|_{\  1 \  jk}^{\  jk \  1}}+ 
\
\
\displaystyle
{|_{\  i \  jk}^{\  jk \  i}}+{|_{\  j \  jk}^{\  jk \  j}}+{|_{\  k \  jk}^{\  jk \  k}}+{|_{\  ij \  jk}^{\  jk \  ij}}+ 
\
\
\displaystyle
{|_{\  ik \  jk}^{\  jk \  ik}}+{|_{\  jk \  jk}^{\  jk \  jk}}+{|_{\  ijk \  jk}^{\  jk \  ijk}}+ 
\
\
\displaystyle
{|_{\  1 \  ijk}^{\  ijk \  1}}+{|_{\  i \  ijk}^{\  ijk \  i}}+{|_{\  j \  ijk}^{\  ijk \  j}}+{|_{\  k \  ijk}^{\  ijk \  k}}+ 
\
\
\displaystyle
{|_{\  ij \  ijk}^{\  ijk \  ij}}+{|_{\  ik \  ijk}^{\  ijk \  ik}}+{|_{\  jk \  ijk}^{\  ijk \  jk}}+ 
\
\
\displaystyle
{|_{\  ijk \  ijk}^{\  ijk \  ijk}}
(6)
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
V:𝐋:=ev(1) -- evaluation

\label{eq7}{|^{\  1 \  1}}+{|^{\  i \  i}}+{|^{\  j \  j}}+{|^{\  k \  k}}+{|^{\  ij \  ij}}+{|^{\  ik \  ik}}+{|^{\  jk \  jk}}+{|^{\  ijk \  ijk}}(7)
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
Λ:𝐋:=co(1) -- co-evaluation

\label{eq8}{|_{\  1 \  1}}+{|_{\  i \  i}}+{|_{\  j \  j}}+{|_{\  k \  k}}+{|_{\  ij \  ij}}+{|_{\  ik \  ik}}+{|_{\  jk \  jk}}+{|_{\  ijk \  ijk}}(8)
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
equate(eq)==map((x,y)+->(x=y),ravel lhs eq, ravel rhs eq);
Type: Void

Now generate structure constants for Pauli Algebra

The basis consists of the real and imaginary units. We use quaternion multiplication to form the "multiplication table" as a matrix. Then the structure constants can be obtained by dividing each matrix entry by the list of basis vectors.

The Pauli Algebra as Cl(3)

Basis: Each B.i is a Clifford number

fricas
q0:=sp('i,[2])

\label{eq9}i^{2}(9)
Type: Symbol
fricas
q1:=sp('j,[2])

\label{eq10}j^{2}(10)
Type: Symbol
fricas
q2:=sp('k,[2])

\label{eq11}k^{2}(11)
Type: Symbol
fricas
QQ:=CliffordAlgebra(3,ℚ,matrix [[q0,0,0],[0,q1,0],[0,0,q2]])

\label{eq12}\hbox{\axiomType{CliffordAlgebra}\ } (3, \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , [ [ i [ ; 2 ] , 0, 0 ] , [ 0, j [ ; 2 ] , 0 ] , [ 0, 0, k [ ; 2 ] ] ])(12)
Type: Type
fricas
B:ℒ QQ := [monomial(1,[]),monomial(1,[1]),monomial(1,[2]),monomial(1,[3]),monomial(1,[1,2]),monomial(1,[1,3]),monomial(1,[2,3]),monomial(1,[1,2,3])]

\label{eq13}\left[ 1, \:{e_{1}}, \:{e_{2}}, \:{e_{3}}, \:{{e_{1}}\ {e_{2}}}, \:{{e_{1}}\ {e_{3}}}, \:{{e_{2}}\ {e_{3}}}, \:{{e_{1}}\ {e_{2}}\ {e_{3}}}\right](13)
Type: List(CliffordAlgebra?(3,Expression(Integer),[[i[;2],0,0],[0,j[;2],0],[0,0,k[;2]]]))
fricas
M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j, i,1..dim), j,1..dim)

\label{eq14}\left[ 
\begin{array}{cccccccc}
1 &{e_{1}}&{e_{2}}&{e_{3}}&{{e_{1}}\ {e_{2}}}&{{e_{1}}\ {e_{3}}}&{{e_{2}}\ {e_{3}}}&{{e_{1}}\ {e_{2}}\ {e_{3}}}
\
{e_{1}}&{i^{2}}& -{{e_{1}}\ {e_{2}}}& -{{e_{1}}\ {e_{3}}}& -{{i^{2}}\ {e_{2}}}& -{{i^{2}}\ {e_{3}}}&{{e_{1}}\ {e_{2}}\ {e_{3}}}&{{i^{2}}\ {e_{2}}\ {e_{3}}}
\
{e_{2}}&{{e_{1}}\ {e_{2}}}&{j^{2}}& -{{e_{2}}\ {e_{3}}}&{{j^{2}}\ {e_{1}}}& -{{e_{1}}\ {e_{2}}\ {e_{3}}}& -{{j^{2}}\ {e_{3}}}& -{{j^{2}}\ {e_{1}}\ {e_{3}}}
\
{e_{3}}&{{e_{1}}\ {e_{3}}}&{{e_{2}}\ {e_{3}}}&{k^{2}}&{{e_{1}}\ {e_{2}}\ {e_{3}}}&{{k^{2}}\ {e_{1}}}&{{k^{2}}\ {e_{2}}}&{{k^{2}}\ {e_{1}}\ {e_{2}}}
\
{{e_{1}}\ {e_{2}}}&{{i^{2}}\ {e_{2}}}& -{{j^{2}}\ {e_{1}}}&{{e_{1}}\ {e_{2}}\ {e_{3}}}& -{{i^{2}}\ {j^{2}}}&{{i^{2}}\ {e_{2}}\ {e_{3}}}& -{{j^{2}}\ {e_{1}}\ {e_{3}}}& -{{i^{2}}\ {j^{2}}\ {e_{3}}}
\
{{e_{1}}\ {e_{3}}}&{{i^{2}}\ {e_{3}}}& -{{e_{1}}\ {e_{2}}\ {e_{3}}}& -{{k^{2}}\ {e_{1}}}& -{{i^{2}}\ {e_{2}}\ {e_{3}}}& -{{i^{2}}\ {k^{2}}}&{{k^{2}}\ {e_{1}}\ {e_{2}}}&{{i^{2}}\ {k^{2}}\ {e_{2}}}
\
{{e_{2}}\ {e_{3}}}&{{e_{1}}\ {e_{2}}\ {e_{3}}}&{{j^{2}}\ {e_{3}}}& -{{k^{2}}\ {e_{2}}}&{{j^{2}}\ {e_{1}}\ {e_{3}}}& -{{k^{2}}\ {e_{1}}\ {e_{2}}}& -{{j^{2}}\ {k^{2}}}& -{{j^{2}}\ {k^{2}}\ {e_{1}}}
\
{{e_{1}}\ {e_{2}}\ {e_{3}}}&{{i^{2}}\ {e_{2}}\ {e_{3}}}& -{{j^{2}}\ {e_{1}}\ {e_{3}}}&{{k^{2}}\ {e_{1}}\ {e_{2}}}& -{{i^{2}}\ {j^{2}}\ {e_{3}}}&{{i^{2}}\ {k^{2}}\ {e_{2}}}& -{{j^{2}}\ {k^{2}}\ {e_{1}}}& -{{i^{2}}\ {j^{2}}\ {k^{2}}}
(14)
Type: Matrix(CliffordAlgebra?(3,Expression(Integer),[[i[;2],0,0],[0,j[;2],0],[0,0,k[;2]]]))
fricas
S(y) == map(x +-> coefficient(recip(y)*x,[]),M)
Type: Void
fricas
ѕ :=map(S,B)::ℒ ℒ ℒ ℚ
fricas
Compiling function S with type CliffordAlgebra(3,Expression(Integer)
      ,[[i[;2],0,0],[0,j[;2],0],[0,0,k[;2]]]) -> Matrix(Expression(
      Integer))

\label{eq15}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
\left[{\left[ 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \:{i^{2}}, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: \right.
\
\
\displaystyle
\left.{\left[ 0, \: 0, \:{j^{2}}, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \:{k^{2}}, \: 0, \: 0, \: 0, \: 0 \right]}, \: \right.
\
\
\displaystyle
\left.{\left[ 0, \: 0, \: 0, \: 0, \: -{{i^{2}}\ {j^{2}}}, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \: 0, \: 0, \: -{{i^{2}}\ {k^{2}}}, \: 0, \: 0 \right]}, \: \right.
\
\
\displaystyle
\left.{\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: -{{j^{2}}\ {k^{2}}}, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: -{{i^{2}}\ {j^{2}}\ {k^{2}}}\right]}\right] 
(15)
Type: List(List(List(Expression(Integer))))
fricas
-- structure constants form a tensor operator
Y := Σ(Σ(Σ(ѕ(i)(k)(j)*𝐞.i*𝐝.j*𝐝.k, i,1..dim), j,1..dim), k,1..dim)

\label{eq16}\begin{array}{@{}l}
\displaystyle
{|_{\  1}^{\  1 \  1}}+{|_{\  i}^{\  1 \  i}}+{|_{\  j}^{\  1 \  j}}+{|_{\  k}^{\  1 \  k}}+{|_{\  ij}^{\  1 \  ij}}+{|_{\  ik}^{\  1 \  ik}}+ 
\
\
\displaystyle
{|_{\  jk}^{\  1 \  jk}}+{|_{\  ijk}^{\  1 \  ijk}}+{|_{\  i}^{\  i \  1}}+{{i^{2}}\ {|_{\  1}^{\  i \  i}}}+{|_{\  ij}^{\  i \  j}}+ 
\
\
\displaystyle
{|_{\  ik}^{\  i \  k}}+{{i^{2}}\ {|_{\  j}^{\  i \  ij}}}+{{i^{2}}\ {|_{\  k}^{\  i \  ik}}}+{|_{\  ijk}^{\  i \  jk}}+{{i^{2}}\ {|_{\  jk}^{\  i \  ijk}}}+ 
\
\
\displaystyle
{|_{\  j}^{\  j \  1}}-{|_{\  ij}^{\  j \  i}}+{{j^{2}}\ {|_{\  1}^{\  j \  j}}}+{|_{\  jk}^{\  j \  k}}-{{j^{2}}\ {|_{\  i}^{\  j \  ij}}}- 
\
\
\displaystyle
{|_{\  ijk}^{\  j \  ik}}+{{j^{2}}\ {|_{\  k}^{\  j \  jk}}}-{{j^{2}}\ {|_{\  ik}^{\  j \  ijk}}}+{|_{\  k}^{\  k \  1}}-{|_{\  ik}^{\  k \  i}}- 
\
\
\displaystyle
{|_{\  jk}^{\  k \  j}}+{{k^{2}}\ {|_{\  1}^{\  k \  k}}}+{|_{\  ijk}^{\  k \  ij}}-{{k^{2}}\ {|_{\  i}^{\  k \  ik}}}-{{k^{2}}\ {|_{\  j}^{\  k \  jk}}}+ 
\
\
\displaystyle
{{k^{2}}\ {|_{\  ij}^{\  k \  ijk}}}+{|_{\  ij}^{\  ij \  1}}-{{i^{2}}\ {|_{\  j}^{\  ij \  i}}}+{{j^{2}}\ {|_{\  i}^{\  ij \  j}}}+{|_{\  ijk}^{\  ij \  k}}- 
\
\
\displaystyle
{{i^{2}}\ {j^{2}}\ {|_{\  1}^{\  ij \  ij}}}-{{i^{2}}\ {|_{\  jk}^{\  ij \  ik}}}+{{j^{2}}\ {|_{\  ik}^{\  ij \  jk}}}- 
\
\
\displaystyle
{{i^{2}}\ {j^{2}}\ {|_{\  k}^{\  ij \  ijk}}}+{|_{\  ik}^{\  ik \  1}}-{{i^{2}}\ {|_{\  k}^{\  ik \  i}}}-{|_{\  ijk}^{\  ik \  j}}+ 
\
\
\displaystyle
{{k^{2}}\ {|_{\  i}^{\  ik \  k}}}+{{i^{2}}\ {|_{\  jk}^{\  ik \  ij}}}-{{i^{2}}\ {k^{2}}\ {|_{\  1}^{\  ik \  ik}}}-{{k^{2}}\ {|_{\  ij}^{\  ik \  jk}}}+ 
\
\
\displaystyle
{{i^{2}}\ {k^{2}}\ {|_{\  j}^{\  ik \  ijk}}}+{|_{\  jk}^{\  jk \  1}}+{|_{\  ijk}^{\  jk \  i}}-{{j^{2}}\ {|_{\  k}^{\  jk \  j}}}+ 
\
\
\displaystyle
{{k^{2}}\ {|_{\  j}^{\  jk \  k}}}-{{j^{2}}\ {|_{\  ik}^{\  jk \  ij}}}+{{k^{2}}\ {|_{\  ij}^{\  jk \  ik}}}-{{j^{2}}\ {k^{2}}\ {|_{\  1}^{\  jk \  jk}}}- 
\
\
\displaystyle
{{j^{2}}\ {k^{2}}\ {|_{\  i}^{\  jk \  ijk}}}+{|_{\  ijk}^{\  ijk \  1}}+{{i^{2}}\ {|_{\  jk}^{\  ijk \  i}}}-{{j^{2}}\ {|_{\  ik}^{\  ijk \  j}}}+ 
\
\
\displaystyle
{{k^{2}}\ {|_{\  ij}^{\  ijk \  k}}}-{{i^{2}}\ {j^{2}}\ {|_{\  k}^{\  ijk \  ij}}}+{{i^{2}}\ {k^{2}}\ {|_{\  j}^{\  ijk \  ik}}}- 
\
\
\displaystyle
{{j^{2}}\ {k^{2}}\ {|_{\  i}^{\  ijk \  jk}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {|_{\  1}^{\  ijk \  ijk}}}
(16)
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y, i,1..dim), j,1..dim)

\label{eq17}\left[ 
\begin{array}{cccccccc}
{|_{\  1}}&{|_{\  i}}&{|_{\  j}}&{|_{\  k}}&{|_{\  ij}}&{|_{\  ik}}&{|_{\  jk}}&{|_{\  ijk}}
\
{|_{\  i}}&{{i^{2}}\ {|_{\  1}}}& -{|_{\  ij}}& -{|_{\  ik}}& -{{i^{2}}\ {|_{\  j}}}& -{{i^{2}}\ {|_{\  k}}}&{|_{\  ijk}}&{{i^{2}}\ {|_{\  jk}}}
\
{|_{\  j}}&{|_{\  ij}}&{{j^{2}}\ {|_{\  1}}}& -{|_{\  jk}}&{{j^{2}}\ {|_{\  i}}}& -{|_{\  ijk}}& -{{j^{2}}\ {|_{\  k}}}& -{{j^{2}}\ {|_{\  ik}}}
\
{|_{\  k}}&{|_{\  ik}}&{|_{\  jk}}&{{k^{2}}\ {|_{\  1}}}&{|_{\  ijk}}&{{k^{2}}\ {|_{\  i}}}&{{k^{2}}\ {|_{\  j}}}&{{k^{2}}\ {|_{\  ij}}}
\
{|_{\  ij}}&{{i^{2}}\ {|_{\  j}}}& -{{j^{2}}\ {|_{\  i}}}&{|_{\  ijk}}& -{{i^{2}}\ {j^{2}}\ {|_{\  1}}}&{{i^{2}}\ {|_{\  jk}}}& -{{j^{2}}\ {|_{\  ik}}}& -{{i^{2}}\ {j^{2}}\ {|_{\  k}}}
\
{|_{\  ik}}&{{i^{2}}\ {|_{\  k}}}& -{|_{\  ijk}}& -{{k^{2}}\ {|_{\  i}}}& -{{i^{2}}\ {|_{\  jk}}}& -{{i^{2}}\ {k^{2}}\ {|_{\  1}}}&{{k^{2}}\ {|_{\  ij}}}&{{i^{2}}\ {k^{2}}\ {|_{\  j}}}
\
{|_{\  jk}}&{|_{\  ijk}}&{{j^{2}}\ {|_{\  k}}}& -{{k^{2}}\ {|_{\  j}}}&{{j^{2}}\ {|_{\  ik}}}& -{{k^{2}}\ {|_{\  ij}}}& -{{j^{2}}\ {k^{2}}\ {|_{\  1}}}& -{{j^{2}}\ {k^{2}}\ {|_{\  i}}}
\
{|_{\  ijk}}&{{i^{2}}\ {|_{\  jk}}}& -{{j^{2}}\ {|_{\  ik}}}&{{k^{2}}\ {|_{\  ij}}}& -{{i^{2}}\ {j^{2}}\ {|_{\  k}}}&{{i^{2}}\ {k^{2}}\ {|_{\  j}}}& -{{j^{2}}\ {k^{2}}\ {|_{\  i}}}& -{{i^{2}}\ {j^{2}}\ {k^{2}}\ {|_{\  1}}}
(17)
Type: Matrix(ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)))
fricas
XY := X/Y;
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

Units

fricas
e:=𝐞.1; i:=𝐞.2; j:=𝐞.3; k:=𝐞.4; ij:=𝐞.5; ik:=𝐞.6; jk:=𝐞.7; ijk:=𝐞.8;
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

Multiplication of arbitrary quaternions a and b

fricas
a:=Σ(sb('a,[i])*𝐞.i, i,1..dim)

\label{eq18}\begin{array}{@{}l}
\displaystyle
{{a_{1}}\ {|_{\  1}}}+{{a_{2}}\ {|_{\  i}}}+{{a_{3}}\ {|_{\  j}}}+{{a_{4}}\ {|_{\  k}}}+{{a_{5}}\ {|_{\  ij}}}+{{a_{6}}\ {|_{\  ik}}}+{{a_{7}}\ {|_{\  jk}}}+ 
\
\
\displaystyle
{{a_{8}}\ {|_{\  ijk}}}
(18)
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
b:=Σ(sb('b,[i])*𝐞.i, i,1..dim)

\label{eq19}\begin{array}{@{}l}
\displaystyle
{{b_{1}}\ {|_{\  1}}}+{{b_{2}}\ {|_{\  i}}}+{{b_{3}}\ {|_{\  j}}}+{{b_{4}}\ {|_{\  k}}}+{{b_{5}}\ {|_{\  ij}}}+{{b_{6}}\ {|_{\  ik}}}+{{b_{7}}\ {|_{\  jk}}}+ 
\
\
\displaystyle
{{b_{8}}\ {|_{\  ijk}}}
(19)
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
(a*b)/Y

\label{eq20}\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {a_{8}}\ {b_{8}}}-{{j^{2}}\ {k^{2}}\ {a_{7}}\ {b_{7}}}-{{i^{2}}\ {k^{2}}\ {a_{6}}\ {b_{6}}}- 
\
\
\displaystyle
{{i^{2}}\ {j^{2}}\ {a_{5}}\ {b_{5}}}+{{k^{2}}\ {a_{4}}\ {b_{4}}}+{{j^{2}}\ {a_{3}}\ {b_{3}}}+{{i^{2}}\ {a_{2}}\ {b_{2}}}+ 
\
\
\displaystyle
{{a_{1}}\ {b_{1}}}
(20)
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

Multiplication is Associative

fricas
test(
  ( I Y ) / _
  (  Y  ) = _
  ( Y I ) / _
  (  Y  ) )

\label{eq21} \mbox{\rm true} (21)
Type: Boolean

A scalar product is denoted by the (2,0)-tensor U = \{ u_{ij} \}

fricas
U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)

\label{eq22}\begin{array}{@{}l}
\displaystyle
{{u^{1, \: 1}}\ {|^{\  1 \  1}}}+{{u^{1, \: 2}}\ {|^{\  1 \  i}}}+{{u^{1, \: 3}}\ {|^{\  1 \  j}}}+{{u^{1, \: 4}}\ {|^{\  1 \  k}}}+ 
\
\
\displaystyle
{{u^{1, \: 5}}\ {|^{\  1 \  ij}}}+{{u^{1, \: 6}}\ {|^{\  1 \  ik}}}+{{u^{1, \: 7}}\ {|^{\  1 \  jk}}}+{{u^{1, \: 8}}\ {|^{\  1 \  ijk}}}+ 
\
\
\displaystyle
{{u^{2, \: 1}}\ {|^{\  i \  1}}}+{{u^{2, \: 2}}\ {|^{\  i \  i}}}+{{u^{2, \: 3}}\ {|^{\  i \  j}}}+{{u^{2, \: 4}}\ {|^{\  i \  k}}}+{{u^{2, \: 5}}\ {|^{\  i \  ij}}}+ 
\
\
\displaystyle
{{u^{2, \: 6}}\ {|^{\  i \  ik}}}+{{u^{2, \: 7}}\ {|^{\  i \  jk}}}+{{u^{2, \: 8}}\ {|^{\  i \  ijk}}}+{{u^{3, \: 1}}\ {|^{\  j \  1}}}+ 
\
\
\displaystyle
{{u^{3, \: 2}}\ {|^{\  j \  i}}}+{{u^{3, \: 3}}\ {|^{\  j \  j}}}+{{u^{3, \: 4}}\ {|^{\  j \  k}}}+{{u^{3, \: 5}}\ {|^{\  j \  ij}}}+ 
\
\
\displaystyle
{{u^{3, \: 6}}\ {|^{\  j \  ik}}}+{{u^{3, \: 7}}\ {|^{\  j \  jk}}}+{{u^{3, \: 8}}\ {|^{\  j \  ijk}}}+{{u^{4, \: 1}}\ {|^{\  k \  1}}}+ 
\
\
\displaystyle
{{u^{4, \: 2}}\ {|^{\  k \  i}}}+{{u^{4, \: 3}}\ {|^{\  k \  j}}}+{{u^{4, \: 4}}\ {|^{\  k \  k}}}+{{u^{4, \: 5}}\ {|^{\  k \  ij}}}+ 
\
\
\displaystyle
{{u^{4, \: 6}}\ {|^{\  k \  ik}}}+{{u^{4, \: 7}}\ {|^{\  k \  jk}}}+{{u^{4, \: 8}}\ {|^{\  k \  ijk}}}+{{u^{5, \: 1}}\ {|^{\  ij \  1}}}+ 
\
\
\displaystyle
{{u^{5, \: 2}}\ {|^{\  ij \  i}}}+{{u^{5, \: 3}}\ {|^{\  ij \  j}}}+{{u^{5, \: 4}}\ {|^{\  ij \  k}}}+{{u^{5, \: 5}}\ {|^{\  ij \  ij}}}+ 
\
\
\displaystyle
{{u^{5, \: 6}}\ {|^{\  ij \  ik}}}+{{u^{5, \: 7}}\ {|^{\  ij \  jk}}}+{{u^{5, \: 8}}\ {|^{\  ij \  ijk}}}+{{u^{6, \: 1}}\ {|^{\  ik \  1}}}+ 
\
\
\displaystyle
{{u^{6, \: 2}}\ {|^{\  ik \  i}}}+{{u^{6, \: 3}}\ {|^{\  ik \  j}}}+{{u^{6, \: 4}}\ {|^{\  ik \  k}}}+{{u^{6, \: 5}}\ {|^{\  ik \  ij}}}+ 
\
\
\displaystyle
{{u^{6, \: 6}}\ {|^{\  ik \  ik}}}+{{u^{6, \: 7}}\ {|^{\  ik \  jk}}}+{{u^{6, \: 8}}\ {|^{\  ik \  ijk}}}+{{u^{7, \: 1}}\ {|^{\  jk \  1}}}+ 
\
\
\displaystyle
{{u^{7, \: 2}}\ {|^{\  jk \  i}}}+{{u^{7, \: 3}}\ {|^{\  jk \  j}}}+{{u^{7, \: 4}}\ {|^{\  jk \  k}}}+{{u^{7, \: 5}}\ {|^{\  jk \  ij}}}+ 
\
\
\displaystyle
{{u^{7, \: 6}}\ {|^{\  jk \  ik}}}+{{u^{7, \: 7}}\ {|^{\  jk \  jk}}}+{{u^{7, \: 8}}\ {|^{\  jk \  ijk}}}+{{u^{8, \: 1}}\ {|^{\  ijk \  1}}}+ 
\
\
\displaystyle
{{u^{8, \: 2}}\ {|^{\  ijk \  i}}}+{{u^{8, \: 3}}\ {|^{\  ijk \  j}}}+{{u^{8, \: 4}}\ {|^{\  ijk \  k}}}+{{u^{8, \: 5}}\ {|^{\  ijk \  ij}}}+ 
\
\
\displaystyle
{{u^{8, \: 6}}\ {|^{\  ijk \  ik}}}+{{u^{8, \: 7}}\ {|^{\  ijk \  jk}}}+{{u^{8, \: 8}}\ {|^{\  ijk \  ijk}}}
(22)
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

Definition 1

We say that the scalar product is associative if the tensor equation holds:

    Y   =   Y
     U     U

In other words, if the (3,0)-tensor:


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\label{eq23}
  \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}
  (23)
(three-point function) is zero.

Using the LinearOperator domain in Axiom and some carefully chosen symbols we can easily enter expressions that are both readable and interpreted by Axiom as "graphical calculus" diagrams describing complex products and compositions of linear operators.

fricas
ω:𝐋 :=                 _
     (    Y I    )  /  _
           U        -  _
     (    I Y    )  /  _
           U;
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

Definition 2

An algebra with a non-degenerate associative scalar product is called a [Frobenius Algebra]?.

The Cartan-Killing Trace

fricas
Ú:=
   (  Y Λ  ) / _
   (   Y I ) / _
        V

\label{eq24}\begin{array}{@{}l}
\displaystyle
{8 \ {|^{\  1 \  1}}}+{8 \ {i^{2}}\ {|^{\  i \  i}}}+{8 \ {j^{2}}\ {|^{\  j \  j}}}+{8 \ {k^{2}}\ {|^{\  k \  k}}}-{8 \ {i^{2}}\ {j^{2}}\ {|^{\  ij \  ij}}}- 
\
\
\displaystyle
{8 \ {i^{2}}\ {k^{2}}\ {|^{\  ik \  ik}}}-{8 \ {j^{2}}\ {k^{2}}\ {|^{\  jk \  jk}}}-{8 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ {|^{\  ijk \  ijk}}}
(24)
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
Ù:=
   (  Λ Y  ) / _
   ( I Y   ) / _
      V

\label{eq25}\begin{array}{@{}l}
\displaystyle
{8 \ {|^{\  1 \  1}}}+{8 \ {i^{2}}\ {|^{\  i \  i}}}+{8 \ {j^{2}}\ {|^{\  j \  j}}}+{8 \ {k^{2}}\ {|^{\  k \  k}}}-{8 \ {i^{2}}\ {j^{2}}\ {|^{\  ij \  ij}}}- 
\
\
\displaystyle
{8 \ {i^{2}}\ {k^{2}}\ {|^{\  ik \  ik}}}-{8 \ {j^{2}}\ {k^{2}}\ {|^{\  jk \  jk}}}-{8 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ {|^{\  ijk \  ijk}}}
(25)
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
test(Ù=Ú)

\label{eq26} \mbox{\rm true} (26)
Type: Boolean

forms a non-degenerate associative scalar product for Y

fricas
Ũ := Ù

\label{eq27}\begin{array}{@{}l}
\displaystyle
{8 \ {|^{\  1 \  1}}}+{8 \ {i^{2}}\ {|^{\  i \  i}}}+{8 \ {j^{2}}\ {|^{\  j \  j}}}+{8 \ {k^{2}}\ {|^{\  k \  k}}}-{8 \ {i^{2}}\ {j^{2}}\ {|^{\  ij \  ij}}}- 
\
\
\displaystyle
{8 \ {i^{2}}\ {k^{2}}\ {|^{\  ik \  ik}}}-{8 \ {j^{2}}\ {k^{2}}\ {|^{\  jk \  jk}}}-{8 \ {i^{2}}\ {j^{2}}\ {k^{2}}\ {|^{\  ijk \  ijk}}}
(27)
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
test
     (    Y I    )  /
           Ũ        =
     (    I Y    )  /
           Ũ

\label{eq28} \mbox{\rm true} (28)
Type: Boolean
fricas
determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ), j,1..dim), i,1..dim)

\label{eq29}{16777216}\ {{i^{2}}^{4}}\ {{j^{2}}^{4}}\ {{k^{2}}^{4}}(29)
Type: Expression(Integer)

General Solution

Frobenius Form (co-unit)

fricas
d:=ε1*𝐝.1+εi*𝐝.2+εj*𝐝.3+εk*𝐝.4+εij*𝐝.5+εik*𝐝.6+εjk*𝐝.7+εijk*𝐝.8

\label{eq30}{�� 1 \ {|^{\  1}}}+{�� i \ {|^{\  i}}}+{�� j \ {|^{\  j}}}+{�� k \ {|^{\  k}}}+{�� ij \ {|^{\  ij}}}+{�� ik \ {|^{\  ik}}}+{�� jk \ {|^{\  jk}}}+{�� ijk \ {|^{\  ijk}}}(30)
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
Ų:= Y/d

\label{eq31}\begin{array}{@{}l}
\displaystyle
{�� 1 \ {|^{\  1 \  1}}}+{�� i \ {|^{\  1 \  i}}}+{�� j \ {|^{\  1 \  j}}}+{�� k \ {|^{\  1 \  k}}}+{�� ij \ {|^{\  1 \  ij}}}+{�� ik \ {|^{\  1 \  ik}}}+ 
\
\
\displaystyle
{�� jk \ {|^{\  1 \  jk}}}+{�� ijk \ {|^{\  1 \  ijk}}}+{�� i \ {|^{\  i \  1}}}+{{i^{2}}\  �� 1 \ {|^{\  i \  i}}}+{�� ij \ {|^{\  i \  j}}}+ 
\
\
\displaystyle
{�� ik \ {|^{\  i \  k}}}+{{i^{2}}\  �� j \ {|^{\  i \  ij}}}+{{i^{2}}\  �� k \ {|^{\  i \  ik}}}+{�� ijk \ {|^{\  i \  jk}}}+{{i^{2}}\  �� jk \ {|^{\  i \  ijk}}}+ 
\
\
\displaystyle
{�� j \ {|^{\  j \  1}}}-{�� ij \ {|^{\  j \  i}}}+{{j^{2}}\  �� 1 \ {|^{\  j \  j}}}+{�� jk \ {|^{\  j \  k}}}-{{j^{2}}\  �� i \ {|^{\  j \  ij}}}- 
\
\
\displaystyle
{�� ijk \ {|^{\  j \  ik}}}+{{j^{2}}\  �� k \ {|^{\  j \  jk}}}-{{j^{2}}\  �� ik \ {|^{\  j \  ijk}}}+{�� k \ {|^{\  k \  1}}}-{�� ik \ {|^{\  k \  i}}}- 
\
\
\displaystyle
{�� jk \ {|^{\  k \  j}}}+{{k^{2}}\  �� 1 \ {|^{\  k \  k}}}+{�� ijk \ {|^{\  k \  ij}}}-{{k^{2}}\  �� i \ {|^{\  k \  ik}}}-{{k^{2}}\  �� j \ {|^{\  k \  jk}}}+ 
\
\
\displaystyle
{{k^{2}}\  �� ij \ {|^{\  k \  ijk}}}+{�� ij \ {|^{\  ij \  1}}}-{{i^{2}}\  �� j \ {|^{\  ij \  i}}}+{{j^{2}}\  �� i \ {|^{\  ij \  j}}}+{�� ijk \ {|^{\  ij \  k}}}- 
\
\
\displaystyle
{{i^{2}}\ {j^{2}}\  �� 1 \ {|^{\  ij \  ij}}}-{{i^{2}}\  �� jk \ {|^{\  ij \  ik}}}+{{j^{2}}\  �� ik \ {|^{\  ij \  jk}}}- 
\
\
\displaystyle
{{i^{2}}\ {j^{2}}\  �� k \ {|^{\  ij \  ijk}}}+{�� ik \ {|^{\  ik \  1}}}-{{i^{2}}\  �� k \ {|^{\  ik \  i}}}-{�� ijk \ {|^{\  ik \  j}}}+ 
\
\
\displaystyle
{{k^{2}}\  �� i \ {|^{\  ik \  k}}}+{{i^{2}}\  �� jk \ {|^{\  ik \  ij}}}-{{i^{2}}\ {k^{2}}\  �� 1 \ {|^{\  ik \  ik}}}-{{k^{2}}\  �� ij \ {|^{\  ik \  jk}}}+ 
\
\
\displaystyle
{{i^{2}}\ {k^{2}}\  �� j \ {|^{\  ik \  ijk}}}+{�� jk \ {|^{\  jk \  1}}}+{�� ijk \ {|^{\  jk \  i}}}-{{j^{2}}\  �� k \ {|^{\  jk \  j}}}+ 
\
\
\displaystyle
{{k^{2}}\  �� j \ {|^{\  jk \  k}}}-{{j^{2}}\  �� ik \ {|^{\  jk \  ij}}}+{{k^{2}}\  �� ij \ {|^{\  jk \  ik}}}-{{j^{2}}\ {k^{2}}\  �� 1 \ {|^{\  jk \  jk}}}- 
\
\
\displaystyle
{{j^{2}}\ {k^{2}}\  �� i \ {|^{\  jk \  ijk}}}+{�� ijk \ {|^{\  ijk \  1}}}+{{i^{2}}\  �� jk \ {|^{\  ijk \  i}}}-{{j^{2}}\  �� ik \ {|^{\  ijk \  j}}}+ 
\
\
\displaystyle
{{k^{2}}\  �� ij \ {|^{\  ijk \  k}}}-{{i^{2}}\ {j^{2}}\  �� k \ {|^{\  ijk \  ij}}}+{{i^{2}}\ {k^{2}}\  �� j \ {|^{\  ijk \  ik}}}- 
\
\
\displaystyle
{{j^{2}}\ {k^{2}}\  �� i \ {|^{\  ijk \  jk}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \ {|^{\  ijk \  ijk}}}
(31)
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

In general the pairing is not symmetric!

fricas
u1:=matrix Ξ(Ξ(retract((𝐞.i 𝐞.j)/Ų), i,1..dim), j,1..dim)

\label{eq32}\left[ 
\begin{array}{cccccccc}
�� 1 & �� i & �� j & �� k & �� ij & �� ik & �� jk & �� ijk 
\
�� i &{{i^{2}}\  �� 1}& - �� ij & - �� ik & -{{i^{2}}\  �� j}& -{{i^{2}}\  �� k}& �� ijk &{{i^{2}}\  �� jk}
\
�� j & �� ij &{{j^{2}}\  �� 1}& - �� jk &{{j^{2}}\  �� i}& - �� ijk & -{{j^{2}}\  �� k}& -{{j^{2}}\  �� ik}
\
�� k & �� ik & �� jk &{{k^{2}}\  �� 1}& �� ijk &{{k^{2}}\  �� i}&{{k^{2}}\  �� j}&{{k^{2}}\  �� ij}
\
�� ij &{{i^{2}}\  �� j}& -{{j^{2}}\  �� i}& �� ijk & -{{i^{2}}\ {j^{2}}\  �� 1}&{{i^{2}}\  �� jk}& -{{j^{2}}\  �� ik}& -{{i^{2}}\ {j^{2}}\  �� k}
\
�� ik &{{i^{2}}\  �� k}& - �� ijk & -{{k^{2}}\  �� i}& -{{i^{2}}\  �� jk}& -{{i^{2}}\ {k^{2}}\  �� 1}&{{k^{2}}\  �� ij}&{{i^{2}}\ {k^{2}}\  �� j}
\
�� jk & �� ijk &{{j^{2}}\  �� k}& -{{k^{2}}\  �� j}&{{j^{2}}\  �� ik}& -{{k^{2}}\  �� ij}& -{{j^{2}}\ {k^{2}}\  �� 1}& -{{j^{2}}\ {k^{2}}\  �� i}
\
�� ijk &{{i^{2}}\  �� jk}& -{{j^{2}}\  �� ik}&{{k^{2}}\  �� ij}& -{{i^{2}}\ {j^{2}}\  �� k}&{{i^{2}}\ {k^{2}}\  �� j}& -{{j^{2}}\ {k^{2}}\  �� i}& -{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1}
(32)
Type: Matrix(Expression(Integer))

The scalar product must be non-degenerate:

fricas
--Ů:=determinant u1
--factor(numer Ů)/factor(denom Ů)
1

\label{eq33}1(33)
Type: PositiveInteger?

Cartan-Killing is a special case

fricas
ck:=solve(equate(Ũ=Ų),[ε1,εi,εj,εk,εij,εik,εjk,εijk]).1
fricas
Compiling function equate with type Equation(ClosedLinearOperator(
      OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)))
       -> List(Equation(Expression(Integer)))

\label{eq34}\left[{�� 1 = 8}, \:{�� i = 0}, \:{�� j = 0}, \:{�� k = 0}, \:{�� ij = 0}, \:{�� ik = 0}, \:{�� jk = 0}, \:{�� ijk = 0}\right](34)
Type: List(Equation(Expression(Integer)))

Frobenius scalar product of "vector" quaternions a and b

fricas
a:=sb('a,[1])*i+sb('a,[2])*j+sb('a,[3])*k

\label{eq35}{{a_{1}}\ {|_{\  i}}}+{{a_{2}}\ {|_{\  j}}}+{{a_{3}}\ {|_{\  k}}}(35)
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
b:=sb('b,[1])*i+sb('b,[2])*j+sb('b,[3])*k

\label{eq36}{{b_{1}}\ {|_{\  i}}}+{{b_{2}}\ {|_{\  j}}}+{{b_{3}}\ {|_{\  k}}}(36)
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
(a,a)/Ų

\label{eq37}{\left({{k^{2}}\ {{a_{3}}^{2}}}+{{j^{2}}\ {{a_{2}}^{2}}}+{{i^{2}}\ {{a_{1}}^{2}}}\right)}\  �� 1(37)
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
(b,b)/Ų

\label{eq38}{\left({{k^{2}}\ {{b_{3}}^{2}}}+{{j^{2}}\ {{b_{2}}^{2}}}+{{i^{2}}\ {{b_{1}}^{2}}}\right)}\  �� 1(38)
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
(a,b)/Ų

\label{eq39}\begin{array}{@{}l}
\displaystyle
{{\left({{a_{2}}\ {b_{3}}}-{{a_{3}}\ {b_{2}}}\right)}\  �� jk}+{{\left({{a_{1}}\ {b_{3}}}-{{a_{3}}\ {b_{1}}}\right)}\  �� ik}+{{\left({{a_{1}}\ {b_{2}}}-{{a_{2}}\ {b_{1}}}\right)}\  �� ij}+ 
\
\
\displaystyle
{{\left({{k^{2}}\ {a_{3}}\ {b_{3}}}+{{j^{2}}\ {a_{2}}\ {b_{2}}}+{{i^{2}}\ {a_{1}}\ {b_{1}}}\right)}\  �� 1}
(39)
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

Definition 3

Co-scalar product

Solve the Snake Relation as a system of linear equations.

fricas
mU:=inverse matrix Ξ(Ξ(retract((𝐞.i*𝐞.j)/Ų), i,1..dim), j,1..dim);
Type: Union(Matrix(Expression(Integer)),...)
fricas
Ω:=Σ(Σ(mU(i,j)*(𝐞.i*𝐞.j), i,1..dim), j,1..dim);
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
ΩX:=Ω/X;
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

The common demoninator is 1/\sqrt{\mathring{U}}

fricas
--squareFreePart factor denom Ů / squareFreePart factor numer Ů
matrix Ξ(Ξ(numer retract(Ω/(𝐝.i*𝐝.j)), i,1..dim), j,1..dim)

\label{eq40}\left[ 
\begin{array}{cccccccc}
{-{{{i^{2}}^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\  �� 1 \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� ij \  �� ijk \  �� k}+{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \  �� ijk \  �� jk}-{{{i^{2}}^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\  �� 1 \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� ijk \  �� ik \  �� j}+{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\  �� 1 \ {{�� ik}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \ {{�� ijk}^{2}}}+{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\  �� 1 \ {{�� ij}^{2}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {{k^{2}}^{2}}\  �� 1 \ {{�� i}^{2}}}+{{{i^{2}}^{2}}\ {{j^{2}}^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{3}}}}&{{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\  �� i \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� ij \  �� jk \  �� k}+{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \ {{�� jk}^{2}}}+{{\left(-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� ik \  �� j}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ijk}\right)}\  �� jk}+{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\  �� i \ {{�� j}^{2}}}-{{{j^{2}}^{2}}\ {k^{2}}\  �� i \ {{�� ik}^{2}}}+{{j^{2}}\ {k^{2}}\  �� i \ {{�� ijk}^{2}}}-{{j^{2}}\ {{k^{2}}^{2}}\  �� i \ {{�� ij}^{2}}}+{{{j^{2}}^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{3}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}\  �� i}}&{{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\  �� j \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� ij \  �� ik \  �� k}-{{{i^{2}}^{2}}\ {k^{2}}\  �� j \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \  �� ik \  �� jk}+{{{i^{2}}^{2}}\ {{k^{2}}^{2}}\ {{�� j}^{3}}}+{{\left({{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� ik}^{2}}}+{{i^{2}}\ {k^{2}}\ {{�� ijk}^{2}}}-{{i^{2}}\ {{k^{2}}^{2}}\ {{�� ij}^{2}}}+{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{2}}}-{{{i^{2}}^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}}\right)}\  �� j}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ijk \  �� ik}}&{{{{i^{2}}^{2}}\ {{j^{2}}^{2}}\ {{�� k}^{3}}}+{{\left(-{{{i^{2}}^{2}}\ {j^{2}}\ {{�� jk}^{2}}}+{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ {{�� j}^{2}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {{�� ik}^{2}}}+{{i^{2}}\ {j^{2}}\ {{�� ijk}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� ij}^{2}}}+{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{{i^{2}}^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� k}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \  �� ij \  �� jk}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� ij \  �� ik \  �� j}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ij \  �� ijk}}&{{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� ij \ {{�� k}^{2}}}+{{\left({2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \  �� jk}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� ik \  �� j}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ijk}\right)}\  �� k}+{{i^{2}}\ {k^{2}}\  �� ij \ {{�� jk}^{2}}}-{{i^{2}}\ {{k^{2}}^{2}}\  �� ij \ {{�� j}^{2}}}+{{j^{2}}\ {k^{2}}\  �� ij \ {{�� ik}^{2}}}-{{k^{2}}\  �� ij \ {{�� ijk}^{2}}}+{{{k^{2}}^{2}}\ {{�� ij}^{3}}}+{{\left(-{{j^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}}\right)}\  �� ij}}&{-{{i^{2}}\ {{j^{2}}^{2}}\  �� ik \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� ij \  �� j \  �� k}+{{i^{2}}\ {j^{2}}\  �� ik \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \  �� j \  �� jk}+{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� ik \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ijk \  �� j}+{{{j^{2}}^{2}}\ {{�� ik}^{3}}}+{{\left(-{{j^{2}}\ {{�� ijk}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� ij}^{2}}}-{{{j^{2}}^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� ik}}&{-{{{i^{2}}^{2}}\ {j^{2}}\  �� jk \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \  �� ij \  �� k}+{{{i^{2}}^{2}}\ {{�� jk}^{3}}}+{{\left(-{{{i^{2}}^{2}}\ {k^{2}}\ {{�� j}^{2}}}+{{i^{2}}\ {j^{2}}\ {{�� ik}^{2}}}-{{i^{2}}\ {{�� ijk}^{2}}}+{{i^{2}}\ {k^{2}}\ {{�� ij}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� jk}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \  �� ik \  �� j}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� i \  �� ijk}}&{{{i^{2}}\ {j^{2}}\  �� ijk \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ij \  �� k}-{{i^{2}}\  �� ijk \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� i \  �� jk}+{{i^{2}}\ {k^{2}}\  �� ijk \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ik \  �� j}-{{j^{2}}\  �� ijk \ {{�� ik}^{2}}}+{{�� ijk}^{3}}+{{\left(-{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� ijk}}
\
{{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\  �� i \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� ij \  �� jk \  �� k}+{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \ {{�� jk}^{2}}}+{{\left(-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� ik \  �� j}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ijk}\right)}\  �� jk}+{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\  �� i \ {{�� j}^{2}}}-{{{j^{2}}^{2}}\ {k^{2}}\  �� i \ {{�� ik}^{2}}}+{{j^{2}}\ {k^{2}}\  �� i \ {{�� ijk}^{2}}}-{{j^{2}}\ {{k^{2}}^{2}}\  �� i \ {{�� ij}^{2}}}+{{{j^{2}}^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{3}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}\  �� i}}&{-{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\  �� 1 \ {{�� k}^{2}}}-{2 \ {j^{2}}\ {k^{2}}\  �� ij \  �� ijk \  �� k}+{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \ {{�� jk}^{2}}}-{2 \ {j^{2}}\ {k^{2}}\  �� i \  �� ijk \  �� jk}-{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\  �� 1 \ {{�� j}^{2}}}+{2 \ {j^{2}}\ {k^{2}}\  �� ijk \  �� ik \  �� j}+{{{j^{2}}^{2}}\ {k^{2}}\  �� 1 \ {{�� ik}^{2}}}+{{j^{2}}\ {k^{2}}\  �� 1 \ {{�� ijk}^{2}}}+{{j^{2}}\ {{k^{2}}^{2}}\  �� 1 \ {{�� ij}^{2}}}-{{{j^{2}}^{2}}\ {{k^{2}}^{2}}\  �� 1 \ {{�� i}^{2}}}+{{i^{2}}\ {{j^{2}}^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{3}}}}&{-{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� ij \ {{�� k}^{2}}}+{{\left(-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \  �� jk}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� ik \  �� j}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ijk}\right)}\  �� k}-{{i^{2}}\ {k^{2}}\  �� ij \ {{�� jk}^{2}}}+{{i^{2}}\ {{k^{2}}^{2}}\  �� ij \ {{�� j}^{2}}}-{{j^{2}}\ {k^{2}}\  �� ij \ {{�� ik}^{2}}}+{{k^{2}}\  �� ij \ {{�� ijk}^{2}}}-{{{k^{2}}^{2}}\ {{�� ij}^{3}}}+{{\left({{j^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}}\right)}\  �� ij}}&{{{i^{2}}\ {{j^{2}}^{2}}\  �� ik \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� ij \  �� j \  �� k}-{{i^{2}}\ {j^{2}}\  �� ik \ {{�� jk}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \  �� j \  �� jk}-{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� ik \ {{�� j}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ijk \  �� j}-{{{j^{2}}^{2}}\ {{�� ik}^{3}}}+{{\left({{j^{2}}\ {{�� ijk}^{2}}}-{{j^{2}}\ {k^{2}}\ {{�� ij}^{2}}}+{{{j^{2}}^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� ik}}&{-{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� j \ {{�� k}^{2}}}+{2 \ {j^{2}}\ {k^{2}}\  �� ij \  �� ik \  �� k}+{{i^{2}}\ {k^{2}}\  �� j \ {{�� jk}^{2}}}+{2 \ {j^{2}}\ {k^{2}}\  �� i \  �� ik \  �� jk}-{{i^{2}}\ {{k^{2}}^{2}}\ {{�� j}^{3}}}+{{\left(-{{j^{2}}\ {k^{2}}\ {{�� ik}^{2}}}-{{k^{2}}\ {{�� ijk}^{2}}}+{{{k^{2}}^{2}}\ {{�� ij}^{2}}}-{{j^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}}\right)}\  �� j}-{2 \ {j^{2}}\ {k^{2}}\  �� 1 \  �� ijk \  �� ik}}&{-{{i^{2}}\ {{j^{2}}^{2}}\ {{�� k}^{3}}}+{{\left({{i^{2}}\ {j^{2}}\ {{�� jk}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� j}^{2}}}+{{{j^{2}}^{2}}\ {{�� ik}^{2}}}-{{j^{2}}\ {{�� ijk}^{2}}}-{{j^{2}}\ {k^{2}}\ {{�� ij}^{2}}}-{{{j^{2}}^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� k}-{2 \ {j^{2}}\ {k^{2}}\  �� i \  �� ij \  �� jk}+{2 \ {j^{2}}\ {k^{2}}\  �� ij \  �� ik \  �� j}+{2 \ {j^{2}}\ {k^{2}}\  �� 1 \  �� ij \  �� ijk}}&{{{i^{2}}\ {j^{2}}\  �� ijk \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ij \  �� k}-{{i^{2}}\  �� ijk \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� i \  �� jk}+{{i^{2}}\ {k^{2}}\  �� ijk \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ik \  �� j}-{{j^{2}}\  �� ijk \ {{�� ik}^{2}}}+{{�� ijk}^{3}}+{{\left(-{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� ijk}}&{-{{i^{2}}\ {j^{2}}\  �� jk \ {{�� k}^{2}}}+{2 \ {j^{2}}\ {k^{2}}\  �� i \  �� ij \  �� k}+{{i^{2}}\ {{�� jk}^{3}}}+{{\left(-{{i^{2}}\ {k^{2}}\ {{�� j}^{2}}}+{{j^{2}}\ {{�� ik}^{2}}}-{{�� ijk}^{2}}+{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� jk}-{2 \ {j^{2}}\ {k^{2}}\  �� i \  �� ik \  �� j}-{2 \ {j^{2}}\ {k^{2}}\  �� 1 \  �� i \  �� ijk}}
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{{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\  �� j \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� ij \  �� ik \  �� k}-{{{i^{2}}^{2}}\ {k^{2}}\  �� j \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \  �� ik \  �� jk}+{{{i^{2}}^{2}}\ {{k^{2}}^{2}}\ {{�� j}^{3}}}+{{\left({{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� ik}^{2}}}+{{i^{2}}\ {k^{2}}\ {{�� ijk}^{2}}}-{{i^{2}}\ {{k^{2}}^{2}}\ {{�� ij}^{2}}}+{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{2}}}-{{{i^{2}}^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}}\right)}\  �� j}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ijk \  �� ik}}&{{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� ij \ {{�� k}^{2}}}+{{\left({2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \  �� jk}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� ik \  �� j}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ijk}\right)}\  �� k}+{{i^{2}}\ {k^{2}}\  �� ij \ {{�� jk}^{2}}}-{{i^{2}}\ {{k^{2}}^{2}}\  �� ij \ {{�� j}^{2}}}+{{j^{2}}\ {k^{2}}\  �� ij \ {{�� ik}^{2}}}-{{k^{2}}\  �� ij \ {{�� ijk}^{2}}}+{{{k^{2}}^{2}}\ {{�� ij}^{3}}}+{{\left(-{{j^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}}\right)}\  �� ij}}&{-{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {k^{2}}\  �� ij \  �� ijk \  �� k}+{{{i^{2}}^{2}}\ {k^{2}}\  �� 1 \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {k^{2}}\  �� i \  �� ijk \  �� jk}-{{{i^{2}}^{2}}\ {{k^{2}}^{2}}\  �� 1 \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {k^{2}}\  �� ijk \  �� ik \  �� j}+{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \ {{�� ik}^{2}}}+{{i^{2}}\ {k^{2}}\  �� 1 \ {{�� ijk}^{2}}}+{{i^{2}}\ {{k^{2}}^{2}}\  �� 1 \ {{�� ij}^{2}}}-{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\  �� 1 \ {{�� i}^{2}}}+{{{i^{2}}^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{3}}}}&{{{{i^{2}}^{2}}\ {j^{2}}\  �� jk \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \  �� ij \  �� k}-{{{i^{2}}^{2}}\ {{�� jk}^{3}}}+{{\left({{{i^{2}}^{2}}\ {k^{2}}\ {{�� j}^{2}}}-{{i^{2}}\ {j^{2}}\ {{�� ik}^{2}}}+{{i^{2}}\ {{�� ijk}^{2}}}-{{i^{2}}\ {k^{2}}\ {{�� ij}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� jk}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \  �� ik \  �� j}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� i \  �� ijk}}&{{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {k^{2}}\  �� ij \  �� jk \  �� k}+{{i^{2}}\ {k^{2}}\  �� i \ {{�� jk}^{2}}}+{{\left(-{2 \ {i^{2}}\ {k^{2}}\  �� ik \  �� j}-{2 \ {i^{2}}\ {k^{2}}\  �� 1 \  �� ijk}\right)}\  �� jk}+{{i^{2}}\ {{k^{2}}^{2}}\  �� i \ {{�� j}^{2}}}-{{j^{2}}\ {k^{2}}\  �� i \ {{�� ik}^{2}}}+{{k^{2}}\  �� i \ {{�� ijk}^{2}}}-{{{k^{2}}^{2}}\  �� i \ {{�� ij}^{2}}}+{{j^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{3}}}-{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}\  �� i}}&{-{{i^{2}}\ {j^{2}}\  �� ijk \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ij \  �� k}+{{i^{2}}\  �� ijk \ {{�� jk}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� i \  �� jk}-{{i^{2}}\ {k^{2}}\  �� ijk \ {{�� j}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ik \  �� j}+{{j^{2}}\  �� ijk \ {{�� ik}^{2}}}-{{�� ijk}^{3}}+{{\left({{k^{2}}\ {{�� ij}^{2}}}-{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� ijk}}&{-{{{i^{2}}^{2}}\ {j^{2}}\ {{�� k}^{3}}}+{{\left({{{i^{2}}^{2}}\ {{�� jk}^{2}}}-{{{i^{2}}^{2}}\ {k^{2}}\ {{�� j}^{2}}}+{{i^{2}}\ {j^{2}}\ {{�� ik}^{2}}}-{{i^{2}}\ {{�� ijk}^{2}}}-{{i^{2}}\ {k^{2}}\ {{�� ij}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� k}-{2 \ {i^{2}}\ {k^{2}}\  �� i \  �� ij \  �� jk}+{2 \ {i^{2}}\ {k^{2}}\  �� ij \  �� ik \  �� j}+{2 \ {i^{2}}\ {k^{2}}\  �� 1 \  �� ij \  �� ijk}}&{{{i^{2}}\ {j^{2}}\  �� ik \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {k^{2}}\  �� ij \  �� j \  �� k}-{{i^{2}}\  �� ik \ {{�� jk}^{2}}}+{2 \ {i^{2}}\ {k^{2}}\  �� i \  �� j \  �� jk}-{{i^{2}}\ {k^{2}}\  �� ik \ {{�� j}^{2}}}-{2 \ {i^{2}}\ {k^{2}}\  �� 1 \  �� ijk \  �� j}-{{j^{2}}\ {{�� ik}^{3}}}+{{\left({{�� ijk}^{2}}-{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� ik}}
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{{{{i^{2}}^{2}}\ {{j^{2}}^{2}}\ {{�� k}^{3}}}+{{\left(-{{{i^{2}}^{2}}\ {j^{2}}\ {{�� jk}^{2}}}+{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ {{�� j}^{2}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {{�� ik}^{2}}}+{{i^{2}}\ {j^{2}}\ {{�� ijk}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� ij}^{2}}}+{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{{i^{2}}^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� k}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \  �� ij \  �� jk}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� ij \  �� ik \  �� j}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ij \  �� ijk}}&{-{{i^{2}}\ {{j^{2}}^{2}}\  �� ik \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� ij \  �� j \  �� k}+{{i^{2}}\ {j^{2}}\  �� ik \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \  �� j \  �� jk}+{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� ik \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ijk \  �� j}+{{{j^{2}}^{2}}\ {{�� ik}^{3}}}+{{\left(-{{j^{2}}\ {{�� ijk}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� ij}^{2}}}-{{{j^{2}}^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� ik}}&{-{{{i^{2}}^{2}}\ {j^{2}}\  �� jk \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \  �� ij \  �� k}+{{{i^{2}}^{2}}\ {{�� jk}^{3}}}+{{\left(-{{{i^{2}}^{2}}\ {k^{2}}\ {{�� j}^{2}}}+{{i^{2}}\ {j^{2}}\ {{�� ik}^{2}}}-{{i^{2}}\ {{�� ijk}^{2}}}+{{i^{2}}\ {k^{2}}\ {{�� ij}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� jk}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \  �� ik \  �� j}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� i \  �� ijk}}&{-{{{i^{2}}^{2}}\ {{j^{2}}^{2}}\  �� 1 \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\  �� ij \  �� ijk \  �� k}+{{{i^{2}}^{2}}\ {j^{2}}\  �� 1 \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\  �� i \  �� ijk \  �� jk}-{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\  �� ijk \  �� ik \  �� j}+{{i^{2}}\ {{j^{2}}^{2}}\  �� 1 \ {{�� ik}^{2}}}+{{i^{2}}\ {j^{2}}\  �� 1 \ {{�� ijk}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \ {{�� ij}^{2}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\  �� 1 \ {{�� i}^{2}}}+{{{i^{2}}^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{3}}}}&{{{i^{2}}\ {j^{2}}\  �� ijk \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ij \  �� k}-{{i^{2}}\  �� ijk \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� i \  �� jk}+{{i^{2}}\ {k^{2}}\  �� ijk \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ik \  �� j}-{{j^{2}}\  �� ijk \ {{�� ik}^{2}}}+{{�� ijk}^{3}}+{{\left(-{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� ijk}}&{{{i^{2}}\ {{j^{2}}^{2}}\  �� i \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\  �� ij \  �� jk \  �� k}+{{i^{2}}\ {j^{2}}\  �� i \ {{�� jk}^{2}}}+{{\left(-{2 \ {i^{2}}\ {j^{2}}\  �� ik \  �� j}-{2 \ {i^{2}}\ {j^{2}}\  �� 1 \  �� ijk}\right)}\  �� jk}+{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \ {{�� j}^{2}}}-{{{j^{2}}^{2}}\  �� i \ {{�� ik}^{2}}}+{{j^{2}}\  �� i \ {{�� ijk}^{2}}}-{{j^{2}}\ {k^{2}}\  �� i \ {{�� ij}^{2}}}+{{{j^{2}}^{2}}\ {k^{2}}\ {{�� i}^{3}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{2}}\  �� i}}&{{{{i^{2}}^{2}}\ {j^{2}}\  �� j \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\  �� ij \  �� ik \  �� k}-{{{i^{2}}^{2}}\  �� j \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\  �� i \  �� ik \  �� jk}+{{{i^{2}}^{2}}\ {k^{2}}\ {{�� j}^{3}}}+{{\left({{i^{2}}\ {j^{2}}\ {{�� ik}^{2}}}+{{i^{2}}\ {{�� ijk}^{2}}}-{{i^{2}}\ {k^{2}}\ {{�� ij}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� j}+{2 \ {i^{2}}\ {j^{2}}\  �� 1 \  �� ijk \  �� ik}}&{{{i^{2}}\ {j^{2}}\  �� ij \ {{�� k}^{2}}}+{{\left({2 \ {i^{2}}\ {j^{2}}\  �� i \  �� jk}-{2 \ {i^{2}}\ {j^{2}}\  �� ik \  �� j}-{2 \ {i^{2}}\ {j^{2}}\  �� 1 \  �� ijk}\right)}\  �� k}+{{i^{2}}\  �� ij \ {{�� jk}^{2}}}-{{i^{2}}\ {k^{2}}\  �� ij \ {{�� j}^{2}}}+{{j^{2}}\  �� ij \ {{�� ik}^{2}}}-{�� ij \ {{�� ijk}^{2}}}+{{k^{2}}\ {{�� ij}^{3}}}+{{\left(-{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� ij}}
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{{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� ij \ {{�� k}^{2}}}+{{\left({2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \  �� jk}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� ik \  �� j}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ijk}\right)}\  �� k}+{{i^{2}}\ {k^{2}}\  �� ij \ {{�� jk}^{2}}}-{{i^{2}}\ {{k^{2}}^{2}}\  �� ij \ {{�� j}^{2}}}+{{j^{2}}\ {k^{2}}\  �� ij \ {{�� ik}^{2}}}-{{k^{2}}\  �� ij \ {{�� ijk}^{2}}}+{{{k^{2}}^{2}}\ {{�� ij}^{3}}}+{{\left(-{{j^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}}\right)}\  �� ij}}&{{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� j \ {{�� k}^{2}}}-{2 \ {j^{2}}\ {k^{2}}\  �� ij \  �� ik \  �� k}-{{i^{2}}\ {k^{2}}\  �� j \ {{�� jk}^{2}}}-{2 \ {j^{2}}\ {k^{2}}\  �� i \  �� ik \  �� jk}+{{i^{2}}\ {{k^{2}}^{2}}\ {{�� j}^{3}}}+{{\left({{j^{2}}\ {k^{2}}\ {{�� ik}^{2}}}+{{k^{2}}\ {{�� ijk}^{2}}}-{{{k^{2}}^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}}\right)}\  �� j}+{2 \ {j^{2}}\ {k^{2}}\  �� 1 \  �� ijk \  �� ik}}&{-{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {k^{2}}\  �� ij \  �� jk \  �� k}-{{i^{2}}\ {k^{2}}\  �� i \ {{�� jk}^{2}}}+{{\left({2 \ {i^{2}}\ {k^{2}}\  �� ik \  �� j}+{2 \ {i^{2}}\ {k^{2}}\  �� 1 \  �� ijk}\right)}\  �� jk}-{{i^{2}}\ {{k^{2}}^{2}}\  �� i \ {{�� j}^{2}}}+{{j^{2}}\ {k^{2}}\  �� i \ {{�� ik}^{2}}}-{{k^{2}}\  �� i \ {{�� ijk}^{2}}}+{{{k^{2}}^{2}}\  �� i \ {{�� ij}^{2}}}-{{j^{2}}\ {{k^{2}}^{2}}\ {{�� i}^{3}}}+{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{2}}\  �� i}}&{{{i^{2}}\ {j^{2}}\  �� ijk \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ij \  �� k}-{{i^{2}}\  �� ijk \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� i \  �� jk}+{{i^{2}}\ {k^{2}}\  �� ijk \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ik \  �� j}-{{j^{2}}\  �� ijk \ {{�� ik}^{2}}}+{{�� ijk}^{3}}+{{\left(-{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� ijk}}&{{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \ {{�� k}^{2}}}+{2 \ {k^{2}}\  �� ij \  �� ijk \  �� k}-{{i^{2}}\ {k^{2}}\  �� 1 \ {{�� jk}^{2}}}+{2 \ {k^{2}}\  �� i \  �� ijk \  �� jk}+{{i^{2}}\ {{k^{2}}^{2}}\  �� 1 \ {{�� j}^{2}}}-{2 \ {k^{2}}\  �� ijk \  �� ik \  �� j}-{{j^{2}}\ {k^{2}}\  �� 1 \ {{�� ik}^{2}}}-{{k^{2}}\  �� 1 \ {{�� ijk}^{2}}}-{{{k^{2}}^{2}}\  �� 1 \ {{�� ij}^{2}}}+{{j^{2}}\ {{k^{2}}^{2}}\  �� 1 \ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {{k^{2}}^{2}}\ {{�� 1}^{3}}}}&{-{{i^{2}}\ {j^{2}}\  �� jk \ {{�� k}^{2}}}+{2 \ {j^{2}}\ {k^{2}}\  �� i \  �� ij \  �� k}+{{i^{2}}\ {{�� jk}^{3}}}+{{\left(-{{i^{2}}\ {k^{2}}\ {{�� j}^{2}}}+{{j^{2}}\ {{�� ik}^{2}}}-{{�� ijk}^{2}}+{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� jk}-{2 \ {j^{2}}\ {k^{2}}\  �� i \  �� ik \  �� j}-{2 \ {j^{2}}\ {k^{2}}\  �� 1 \  �� i \  �� ijk}}&{{{i^{2}}\ {j^{2}}\  �� ik \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {k^{2}}\  �� ij \  �� j \  �� k}-{{i^{2}}\  �� ik \ {{�� jk}^{2}}}+{2 \ {i^{2}}\ {k^{2}}\  �� i \  �� j \  �� jk}-{{i^{2}}\ {k^{2}}\  �� ik \ {{�� j}^{2}}}-{2 \ {i^{2}}\ {k^{2}}\  �� 1 \  �� ijk \  �� j}-{{j^{2}}\ {{�� ik}^{3}}}+{{\left({{�� ijk}^{2}}-{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� ik}}&{-{{i^{2}}\ {j^{2}}\ {{�� k}^{3}}}+{{\left({{i^{2}}\ {{�� jk}^{2}}}-{{i^{2}}\ {k^{2}}\ {{�� j}^{2}}}+{{j^{2}}\ {{�� ik}^{2}}}-{{�� ijk}^{2}}-{{k^{2}}\ {{�� ij}^{2}}}-{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� k}-{2 \ {k^{2}}\  �� i \  �� ij \  �� jk}+{2 \ {k^{2}}\  �� ij \  �� ik \  �� j}+{2 \ {k^{2}}\  �� 1 \  �� ij \  �� ijk}}
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{-{{i^{2}}\ {{j^{2}}^{2}}\  �� ik \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� ij \  �� j \  �� k}+{{i^{2}}\ {j^{2}}\  �� ik \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \  �� j \  �� jk}+{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� ik \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ijk \  �� j}+{{{j^{2}}^{2}}\ {{�� ik}^{3}}}+{{\left(-{{j^{2}}\ {{�� ijk}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� ij}^{2}}}-{{{j^{2}}^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� ik}}&{{{i^{2}}\ {{j^{2}}^{2}}\ {{�� k}^{3}}}+{{\left(-{{i^{2}}\ {j^{2}}\ {{�� jk}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� j}^{2}}}-{{{j^{2}}^{2}}\ {{�� ik}^{2}}}+{{j^{2}}\ {{�� ijk}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� ij}^{2}}}+{{{j^{2}}^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� k}+{2 \ {j^{2}}\ {k^{2}}\  �� i \  �� ij \  �� jk}-{2 \ {j^{2}}\ {k^{2}}\  �� ij \  �� ik \  �� j}-{2 \ {j^{2}}\ {k^{2}}\  �� 1 \  �� ij \  �� ijk}}&{-{{i^{2}}\ {j^{2}}\  �� ijk \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ij \  �� k}+{{i^{2}}\  �� ijk \ {{�� jk}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� i \  �� jk}-{{i^{2}}\ {k^{2}}\  �� ijk \ {{�� j}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ik \  �� j}+{{j^{2}}\  �� ijk \ {{�� ik}^{2}}}-{{�� ijk}^{3}}+{{\left({{k^{2}}\ {{�� ij}^{2}}}-{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� ijk}}&{-{{i^{2}}\ {{j^{2}}^{2}}\  �� i \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\  �� ij \  �� jk \  �� k}-{{i^{2}}\ {j^{2}}\  �� i \ {{�� jk}^{2}}}+{{\left({2 \ {i^{2}}\ {j^{2}}\  �� ik \  �� j}+{2 \ {i^{2}}\ {j^{2}}\  �� 1 \  �� ijk}\right)}\  �� jk}-{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \ {{�� j}^{2}}}+{{{j^{2}}^{2}}\  �� i \ {{�� ik}^{2}}}-{{j^{2}}\  �� i \ {{�� ijk}^{2}}}+{{j^{2}}\ {k^{2}}\  �� i \ {{�� ij}^{2}}}-{{{j^{2}}^{2}}\ {k^{2}}\ {{�� i}^{3}}}+{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{2}}\  �� i}}&{{{i^{2}}\ {j^{2}}\  �� jk \ {{�� k}^{2}}}-{2 \ {j^{2}}\ {k^{2}}\  �� i \  �� ij \  �� k}-{{i^{2}}\ {{�� jk}^{3}}}+{{\left({{i^{2}}\ {k^{2}}\ {{�� j}^{2}}}-{{j^{2}}\ {{�� ik}^{2}}}+{{�� ijk}^{2}}-{{k^{2}}\ {{�� ij}^{2}}}-{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� jk}+{2 \ {j^{2}}\ {k^{2}}\  �� i \  �� ik \  �� j}+{2 \ {j^{2}}\ {k^{2}}\  �� 1 \  �� i \  �� ijk}}&{{{i^{2}}\ {{j^{2}}^{2}}\  �� 1 \ {{�� k}^{2}}}+{2 \ {j^{2}}\  �� ij \  �� ijk \  �� k}-{{i^{2}}\ {j^{2}}\  �� 1 \ {{�� jk}^{2}}}+{2 \ {j^{2}}\  �� i \  �� ijk \  �� jk}+{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \ {{�� j}^{2}}}-{2 \ {j^{2}}\  �� ijk \  �� ik \  �� j}-{{{j^{2}}^{2}}\  �� 1 \ {{�� ik}^{2}}}-{{j^{2}}\  �� 1 \ {{�� ijk}^{2}}}-{{j^{2}}\ {k^{2}}\  �� 1 \ {{�� ij}^{2}}}+{{{j^{2}}^{2}}\ {k^{2}}\  �� 1 \ {{�� i}^{2}}}-{{i^{2}}\ {{j^{2}}^{2}}\ {k^{2}}\ {{�� 1}^{3}}}}&{{{i^{2}}\ {j^{2}}\  �� ij \ {{�� k}^{2}}}+{{\left({2 \ {i^{2}}\ {j^{2}}\  �� i \  �� jk}-{2 \ {i^{2}}\ {j^{2}}\  �� ik \  �� j}-{2 \ {i^{2}}\ {j^{2}}\  �� 1 \  �� ijk}\right)}\  �� k}+{{i^{2}}\  �� ij \ {{�� jk}^{2}}}-{{i^{2}}\ {k^{2}}\  �� ij \ {{�� j}^{2}}}+{{j^{2}}\  �� ij \ {{�� ik}^{2}}}-{�� ij \ {{�� ijk}^{2}}}+{{k^{2}}\ {{�� ij}^{3}}}+{{\left(-{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� ij}}&{{{i^{2}}\ {j^{2}}\  �� j \ {{�� k}^{2}}}-{2 \ {j^{2}}\  �� ij \  �� ik \  �� k}-{{i^{2}}\  �� j \ {{�� jk}^{2}}}-{2 \ {j^{2}}\  �� i \  �� ik \  �� jk}+{{i^{2}}\ {k^{2}}\ {{�� j}^{3}}}+{{\left({{j^{2}}\ {{�� ik}^{2}}}+{{�� ijk}^{2}}-{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� j}+{2 \ {j^{2}}\  �� 1 \  �� ijk \  �� ik}}
\
{-{{{i^{2}}^{2}}\ {j^{2}}\  �� jk \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \  �� ij \  �� k}+{{{i^{2}}^{2}}\ {{�� jk}^{3}}}+{{\left(-{{{i^{2}}^{2}}\ {k^{2}}\ {{�� j}^{2}}}+{{i^{2}}\ {j^{2}}\ {{�� ik}^{2}}}-{{i^{2}}\ {{�� ijk}^{2}}}+{{i^{2}}\ {k^{2}}\ {{�� ij}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� jk}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� i \  �� ik \  �� j}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� i \  �� ijk}}&{{{i^{2}}\ {j^{2}}\  �� ijk \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ij \  �� k}-{{i^{2}}\  �� ijk \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� i \  �� jk}+{{i^{2}}\ {k^{2}}\  �� ijk \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ik \  �� j}-{{j^{2}}\  �� ijk \ {{�� ik}^{2}}}+{{�� ijk}^{3}}+{{\left(-{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� ijk}}&{{{{i^{2}}^{2}}\ {j^{2}}\ {{�� k}^{3}}}+{{\left(-{{{i^{2}}^{2}}\ {{�� jk}^{2}}}+{{{i^{2}}^{2}}\ {k^{2}}\ {{�� j}^{2}}}-{{i^{2}}\ {j^{2}}\ {{�� ik}^{2}}}+{{i^{2}}\ {{�� ijk}^{2}}}+{{i^{2}}\ {k^{2}}\ {{�� ij}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� k}+{2 \ {i^{2}}\ {k^{2}}\  �� i \  �� ij \  �� jk}-{2 \ {i^{2}}\ {k^{2}}\  �� ij \  �� ik \  �� j}-{2 \ {i^{2}}\ {k^{2}}\  �� 1 \  �� ij \  �� ijk}}&{-{{{i^{2}}^{2}}\ {j^{2}}\  �� j \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\  �� ij \  �� ik \  �� k}+{{{i^{2}}^{2}}\  �� j \ {{�� jk}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\  �� i \  �� ik \  �� jk}-{{{i^{2}}^{2}}\ {k^{2}}\ {{�� j}^{3}}}+{{\left(-{{i^{2}}\ {j^{2}}\ {{�� ik}^{2}}}-{{i^{2}}\ {{�� ijk}^{2}}}+{{i^{2}}\ {k^{2}}\ {{�� ij}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� j}-{2 \ {i^{2}}\ {j^{2}}\  �� 1 \  �� ijk \  �� ik}}&{-{{i^{2}}\ {j^{2}}\  �� ik \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {k^{2}}\  �� ij \  �� j \  �� k}+{{i^{2}}\  �� ik \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {k^{2}}\  �� i \  �� j \  �� jk}+{{i^{2}}\ {k^{2}}\  �� ik \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {k^{2}}\  �� 1 \  �� ijk \  �� j}+{{j^{2}}\ {{�� ik}^{3}}}+{{\left(-{{�� ijk}^{2}}+{{k^{2}}\ {{�� ij}^{2}}}-{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� ik}}&{-{{i^{2}}\ {j^{2}}\  �� ij \ {{�� k}^{2}}}+{{\left(-{2 \ {i^{2}}\ {j^{2}}\  �� i \  �� jk}+{2 \ {i^{2}}\ {j^{2}}\  �� ik \  �� j}+{2 \ {i^{2}}\ {j^{2}}\  �� 1 \  �� ijk}\right)}\  �� k}-{{i^{2}}\  �� ij \ {{�� jk}^{2}}}+{{i^{2}}\ {k^{2}}\  �� ij \ {{�� j}^{2}}}-{{j^{2}}\  �� ij \ {{�� ik}^{2}}}+{�� ij \ {{�� ijk}^{2}}}-{{k^{2}}\ {{�� ij}^{3}}}+{{\left({{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� ij}}&{{{{i^{2}}^{2}}\ {j^{2}}\  �� 1 \ {{�� k}^{2}}}+{2 \ {i^{2}}\  �� ij \  �� ijk \  �� k}-{{{i^{2}}^{2}}\  �� 1 \ {{�� jk}^{2}}}+{2 \ {i^{2}}\  �� i \  �� ijk \  �� jk}+{{{i^{2}}^{2}}\ {k^{2}}\  �� 1 \ {{�� j}^{2}}}-{2 \ {i^{2}}\  �� ijk \  �� ik \  �� j}-{{i^{2}}\ {j^{2}}\  �� 1 \ {{�� ik}^{2}}}-{{i^{2}}\  �� 1 \ {{�� ijk}^{2}}}-{{i^{2}}\ {k^{2}}\  �� 1 \ {{�� ij}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \ {{�� i}^{2}}}-{{{i^{2}}^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{3}}}}&{-{{i^{2}}\ {j^{2}}\  �� i \ {{�� k}^{2}}}-{2 \ {i^{2}}\  �� ij \  �� jk \  �� k}-{{i^{2}}\  �� i \ {{�� jk}^{2}}}+{{\left({2 \ {i^{2}}\  �� ik \  �� j}+{2 \ {i^{2}}\  �� 1 \  �� ijk}\right)}\  �� jk}-{{i^{2}}\ {k^{2}}\  �� i \ {{�� j}^{2}}}+{{j^{2}}\  �� i \ {{�� ik}^{2}}}-{�� i \ {{�� ijk}^{2}}}+{{k^{2}}\  �� i \ {{�� ij}^{2}}}-{{j^{2}}\ {k^{2}}\ {{�� i}^{3}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}\  �� i}}
\
{{{i^{2}}\ {j^{2}}\  �� ijk \ {{�� k}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ij \  �� k}-{{i^{2}}\  �� ijk \ {{�� jk}^{2}}}-{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� i \  �� jk}+{{i^{2}}\ {k^{2}}\  �� ijk \ {{�� j}^{2}}}+{2 \ {i^{2}}\ {j^{2}}\ {k^{2}}\  �� 1 \  �� ik \  �� j}-{{j^{2}}\  �� ijk \ {{�� ik}^{2}}}+{{�� ijk}^{3}}+{{\left(-{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� ijk}}&{-{{i^{2}}\ {j^{2}}\  �� jk \ {{�� k}^{2}}}+{2 \ {j^{2}}\ {k^{2}}\  �� i \  �� ij \  �� k}+{{i^{2}}\ {{�� jk}^{3}}}+{{\left(-{{i^{2}}\ {k^{2}}\ {{�� j}^{2}}}+{{j^{2}}\ {{�� ik}^{2}}}-{{�� ijk}^{2}}+{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� jk}-{2 \ {j^{2}}\ {k^{2}}\  �� i \  �� ik \  �� j}-{2 \ {j^{2}}\ {k^{2}}\  �� 1 \  �� i \  �� ijk}}&{{{i^{2}}\ {j^{2}}\  �� ik \ {{�� k}^{2}}}+{2 \ {i^{2}}\ {k^{2}}\  �� ij \  �� j \  �� k}-{{i^{2}}\  �� ik \ {{�� jk}^{2}}}+{2 \ {i^{2}}\ {k^{2}}\  �� i \  �� j \  �� jk}-{{i^{2}}\ {k^{2}}\  �� ik \ {{�� j}^{2}}}-{2 \ {i^{2}}\ {k^{2}}\  �� 1 \  �� ijk \  �� j}-{{j^{2}}\ {{�� ik}^{3}}}+{{\left({{�� ijk}^{2}}-{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� ik}}&{{{i^{2}}\ {j^{2}}\  �� ij \ {{�� k}^{2}}}+{{\left({2 \ {i^{2}}\ {j^{2}}\  �� i \  �� jk}-{2 \ {i^{2}}\ {j^{2}}\  �� ik \  �� j}-{2 \ {i^{2}}\ {j^{2}}\  �� 1 \  �� ijk}\right)}\  �� k}+{{i^{2}}\  �� ij \ {{�� jk}^{2}}}-{{i^{2}}\ {k^{2}}\  �� ij \ {{�� j}^{2}}}+{{j^{2}}\  �� ij \ {{�� ik}^{2}}}-{�� ij \ {{�� ijk}^{2}}}+{{k^{2}}\ {{�� ij}^{3}}}+{{\left(-{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� ij}}&{-{{i^{2}}\ {j^{2}}\ {{�� k}^{3}}}+{{\left({{i^{2}}\ {{�� jk}^{2}}}-{{i^{2}}\ {k^{2}}\ {{�� j}^{2}}}+{{j^{2}}\ {{�� ik}^{2}}}-{{�� ijk}^{2}}-{{k^{2}}\ {{�� ij}^{2}}}-{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� k}-{2 \ {k^{2}}\  �� i \  �� ij \  �� jk}+{2 \ {k^{2}}\  �� ij \  �� ik \  �� j}+{2 \ {k^{2}}\  �� 1 \  �� ij \  �� ijk}}&{{{i^{2}}\ {j^{2}}\  �� j \ {{�� k}^{2}}}-{2 \ {j^{2}}\  �� ij \  �� ik \  �� k}-{{i^{2}}\  �� j \ {{�� jk}^{2}}}-{2 \ {j^{2}}\  �� i \  �� ik \  �� jk}+{{i^{2}}\ {k^{2}}\ {{�� j}^{3}}}+{{\left({{j^{2}}\ {{�� ik}^{2}}}+{{�� ijk}^{2}}-{{k^{2}}\ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}}\right)}\  �� j}+{2 \ {j^{2}}\  �� 1 \  �� ijk \  �� ik}}&{-{{i^{2}}\ {j^{2}}\  �� i \ {{�� k}^{2}}}-{2 \ {i^{2}}\  �� ij \  �� jk \  �� k}-{{i^{2}}\  �� i \ {{�� jk}^{2}}}+{{\left({2 \ {i^{2}}\  �� ik \  �� j}+{2 \ {i^{2}}\  �� 1 \  �� ijk}\right)}\  �� jk}-{{i^{2}}\ {k^{2}}\  �� i \ {{�� j}^{2}}}+{{j^{2}}\  �� i \ {{�� ik}^{2}}}-{�� i \ {{�� ijk}^{2}}}+{{k^{2}}\  �� i \ {{�� ij}^{2}}}-{{j^{2}}\ {k^{2}}\ {{�� i}^{3}}}+{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{2}}\  �� i}}&{{{i^{2}}\ {j^{2}}\  �� 1 \ {{�� k}^{2}}}+{2 \  �� ij \  �� ijk \  �� k}-{{i^{2}}\  �� 1 \ {{�� jk}^{2}}}+{2 \  �� i \  �� ijk \  �� jk}+{{i^{2}}\ {k^{2}}\  �� 1 \ {{�� j}^{2}}}-{2 \  �� ijk \  �� ik \  �� j}-{{j^{2}}\  �� 1 \ {{�� ik}^{2}}}-{�� 1 \ {{�� ijk}^{2}}}-{{k^{2}}\  �� 1 \ {{�� ij}^{2}}}+{{j^{2}}\ {k^{2}}\  �� 1 \ {{�� i}^{2}}}-{{i^{2}}\ {j^{2}}\ {k^{2}}\ {{�� 1}^{3}}}}
(40)
Type: Matrix(SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer))))

Check "dimension" and the snake relations.

fricas
O:𝐋:= Ω / Ų

\label{eq41}8(41)
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
test
    (    I ΩX     )  /
    (     Ų I     )  =  I

\label{eq42} \mbox{\rm true} (42)
Type: Boolean
fricas
test
    (     ΩX I    )  /
    (    I Ų      )  =  I

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

Cartan-Killing co-scalar

fricas
eval(Ω,ck)

\label{eq44}\begin{array}{@{}l}
\displaystyle
{{1 \over 8}\ {|_{\  1 \  1}}}+{{1 \over{8 \ {i^{2}}}}\ {|_{\  i \  i}}}+{{1 \over{8 \ {j^{2}}}}\ {|_{\  j \  j}}}+{{1 \over{8 \ {k^{2}}}}\ {|_{\  k \  k}}}- 
\
\
\displaystyle
{{1 \over{8 \ {i^{2}}\ {j^{2}}}}\ {|_{\  ij \  ij}}}-{{1 \over{8 \ {i^{2}}\ {k^{2}}}}\ {|_{\  ik \  ik}}}-{{1 \over{8 \ {j^{2}}\ {k^{2}}}}\ {|_{\  jk \  jk}}}- 
\
\
\displaystyle
{{1 \over{8 \ {i^{2}}\ {j^{2}}\ {k^{2}}}}\ {|_{\  ijk \  ijk}}}
(44)
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

Definition 4

Co-algebra

Compute the "three-point" function and use it to define co-multiplication.

fricas
W:= (Y I) / Ų;
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

fricas
λ:=                      _
     (    I ΩX     )  /  _
     (     Y I     );
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
test
     (     ΩX I    )  /
     (    I  Y     )  =  λ

\label{eq45} \mbox{\rm true} (45)
Type: Boolean

Cartan-Killing co-multiplication

fricas
eval(λ,ck)

\label{eq46}\begin{array}{@{}l}
\displaystyle
{{1 \over 8}\ {|_{\  1 \  1}^{\  1}}}+{{1 \over{8 \ {i^{2}}}}\ {|_{\  i \  i}^{\  1}}}+{{1 \over{8 \ {j^{2}}}}\ {|_{\  j \  j}^{\  1}}}+{{1 \over{8 \ {k^{2}}}}\ {|_{\  k \  k}^{\  1}}}- 
\
\
\displaystyle
{{1 \over{8 \ {i^{2}}\ {j^{2}}}}\ {|_{\  ij \  ij}^{\  1}}}-{{1 \over{8 \ {i^{2}}\ {k^{2}}}}\ {|_{\  ik \  ik}^{\  1}}}-{{1 \over{8 \ {j^{2}}\ {k^{2}}}}\ {|_{\  jk \  jk}^{\  1}}}- 
\
\
\displaystyle
{{1 \over{8 \ {i^{2}}\ {j^{2}}\ {k^{2}}}}\ {|_{\  ijk \  ijk}^{\  1}}}+{{1 \over 8}\ {|_{\  1 \  i}^{\  i}}}+{{1 \over 8}\ {|_{\  i \  1}^{\  i}}}-{{1 \over{8 \ {j^{2}}}}\ {|_{\  j \  ij}^{\  i}}}- 
\
\
\displaystyle
{{1 \over{8 \ {k^{2}}}}\ {|_{\  k \  ik}^{\  i}}}+{{1 \over{8 \ {j^{2}}}}\ {|_{\  ij \  j}^{\  i}}}+{{1 \over{8 \ {k^{2}}}}\ {|_{\  ik \  k}^{\  i}}}- 
\
\
\displaystyle
{{1 \over{8 \ {j^{2}}\ {k^{2}}}}\ {|_{\  jk \  ijk}^{\  i}}}-{{1 \over{8 \ {j^{2}}\ {k^{2}}}}\ {|_{\  ijk \  jk}^{\  i}}}+{{1 \over 8}\ {|_{\  1 \  j}^{\  j}}}+ 
\
\
\displaystyle
{{1 \over{8 \ {i^{2}}}}\ {|_{\  i \  ij}^{\  j}}}+{{1 \over 8}\ {|_{\  j \  1}^{\  j}}}-{{1 \over{8 \ {k^{2}}}}\ {|_{\  k \  jk}^{\  j}}}-{{1 \over{8 \ {i^{2}}}}\ {|_{\  ij \  i}^{\  j}}}+ 
\
\
\displaystyle
{{1 \over{8 \ {i^{2}}\ {k^{2}}}}\ {|_{\  ik \  ijk}^{\  j}}}+{{1 \over{8 \ {k^{2}}}}\ {|_{\  jk \  k}^{\  j}}}+{{1 \over{8 \ {i^{2}}\ {k^{2}}}}\ {|_{\  ijk \  ik}^{\  j}}}+ 
\
\
\displaystyle
{{1 \over 8}\ {|_{\  1 \  k}^{\  k}}}+{{1 \over{8 \ {i^{2}}}}\ {|_{\  i \  ik}^{\  k}}}+{{1 \over{8 \ {j^{2}}}}\ {|_{\  j \  jk}^{\  k}}}+{{1 \over 8}\ {|_{\  k \  1}^{\  k}}}- 
\
\
\displaystyle
{{1 \over{8 \ {i^{2}}\ {j^{2}}}}\ {|_{\  ij \  ijk}^{\  k}}}-{{1 \over{8 \ {i^{2}}}}\ {|_{\  ik \  i}^{\  k}}}-{{1 \over{8 \ {j^{2}}}}\ {|_{\  jk \  j}^{\  k}}}- 
\
\
\displaystyle
{{1 \over{8 \ {i^{2}}\ {j^{2}}}}\ {|_{\  ijk \  ij}^{\  k}}}+{{1 \over 8}\ {|_{\  1 \  ij}^{\  ij}}}+{{1 \over 8}\ {|_{\  i \  j}^{\  ij}}}-{{1 \over 8}\ {|_{\  j \  i}^{\  ij}}}+ 
\
\
\displaystyle
{{1 \over{8 \ {k^{2}}}}\ {|_{\  k \  ijk}^{\  ij}}}+{{1 \over 8}\ {|_{\  ij \  1}^{\  ij}}}-{{1 \over{8 \ {k^{2}}}}\ {|_{\  ik \  jk}^{\  ij}}}+ 
\
\
\displaystyle
{{1 \over{8 \ {k^{2}}}}\ {|_{\  jk \  ik}^{\  ij}}}+{{1 \over{8 \ {k^{2}}}}\ {|_{\  ijk \  k}^{\  ij}}}+{{1 \over 8}\ {|_{\  1 \  ik}^{\  ik}}}+{{1 \over 8}\ {|_{\  i \  k}^{\  ik}}}- 
\
\
\displaystyle
{{1 \over{8 \ {j^{2}}}}\ {|_{\  j \  ijk}^{\  ik}}}-{{1 \over 8}\ {|_{\  k \  i}^{\  ik}}}+{{1 \over{8 \ {j^{2}}}}\ {|_{\  ij \  jk}^{\  ik}}}+{{1 \over 8}\ {|_{\  ik \  1}^{\  ik}}}- \
\
\displaystyle
{{1 \over{8 \ {j^{2}}}}\ {|_{\  jk \  ij}^{\  ik}}}-{{1 \over{8 \ {j^{2}}}}\ {|_{\  ijk \  j}^{\  ik}}}+{{1 \over 8}\ {|_{\  1 \  jk}^{\  jk}}}+ 
\
\
\displaystyle
{{1 \over{8 \ {i^{2}}}}\ {|_{\  i \  ijk}^{\  jk}}}+{{1 \over 8}\ {|_{\  j \  k}^{\  jk}}}-{{1 \over 8}\ {|_{\  k \  j}^{\  jk}}}-{{1 \over{8 \ {i^{2}}}}\ {|_{\  ij \  ik}^{\  jk}}}+ 
\
\
\displaystyle
{{1 \over{8 \ {i^{2}}}}\ {|_{\  ik \  ij}^{\  jk}}}+{{1 \over 8}\ {|_{\  jk \  1}^{\  jk}}}+{{1 \over{8 \ {i^{2}}}}\ {|_{\  ijk \  i}^{\  jk}}}+ 
\
\
\displaystyle
{{1 \over 8}\ {|_{\  1 \  ijk}^{\  ijk}}}+{{1 \over 8}\ {|_{\  i \  jk}^{\  ijk}}}-{{1 \over 8}\ {|_{\  j \  ik}^{\  ijk}}}+{{1 \over 8}\ {|_{\  k \  ij}^{\  ijk}}}+ 
\
\
\displaystyle
{{1 \over 8}\ {|_{\  ij \  k}^{\  ijk}}}-{{1 \over 8}\ {|_{\  ik \  j}^{\  ijk}}}+{{1 \over 8}\ {|_{\  jk \  i}^{\  ijk}}}+{{1 \over 8}\ {|_{\  ijk \  1}^{\  ijk}}}
(46)
Type: ClosedLinearOperator(OrderedVariableList?([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

fricas
test
         e     /
         λ     =    ΩX

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




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