\documentclass{article}$

\usepackage{amsmath,amssymb}$

\sloppy$

\begin{document}$

This output from the file \texttt{program13_25.red}.\\$

Computation of all complex    structures on the real Lie Algebra$

USD {\mathcal{G}}_{6,m5}.USD$



!Case! !U!S!D! xi(2,1)!\neq! 0,! xi(2,3)! =! xi(2,4)! =0,! xi(3,3)=! -xi(2,2),! 
xi(3,4)! !\neq
!\pm! xi(2,1)! !U!S!D.$

\smallskip  \par $

Commutation relations for$

USD {\mathcal{G}}_{6,m5}:USD\\$

USD[x(1),x(3)]=x(5)USD;$

USD[x(1),x(4)]=x(6)USD;$

USD[x(2),x(3)]= - x(6)USD;$

USD[x(2),x(4)]=x(5)USD;$

\P$

Nonzero torsion$

\par$

Torsion equations to cancel (Latex output) : \\USD$

{1,2}|1\\xi(4,2)*xi(1,6) - xi(4,1)*xi(1,5) + xi(3,2)*xi(1,5) + xi(3,1)*xi(1,6)\\
$

{1,2}|2\\xi(4,2)*xi(2,6) - xi(4,1)*xi(2,5) + xi(3,2)*xi(2,5) + xi(3,1)*xi(2,6)\\
$

{1,2}|3\\xi(4,2)*xi(3,6) - xi(4,1)*xi(3,5) + xi(3,6)*xi(3,1) + xi(3,5)*xi(3,2)\\
$

{1,2}|4\\xi(4,6)*xi(4,2) + xi(4,6)*xi(3,1) - xi(4,5)*xi(4,1) + xi(4,5)*xi(3,2)\\
$

{1,2}|5\\xi(5,6)*xi(4,2) + xi(5,6)*xi(3,1) - xi(5,5)*xi(4,1) + xi(5,5)*xi(3,2) -
 xi(4,2)*xi(2,1) + xi(4,1)*xi(2,2) - xi(3,2)*xi(1,1) + xi(3,1)*xi(1,2)\\$

{1,2}|6\\xi(6,6)*xi(4,2) + xi(6,6)*xi(3,1) - xi(6,5)*xi(4,1) + xi(6,5)*xi(3,2) -
 xi(4,2)*xi(1,1) + xi(4,1)*xi(1,2) + xi(3,2)*xi(2,1) - xi(3,1)*xi(2,2)\\$

{1,3}|1\\xi(4,3)*xi(1,6) + xi(3,3)*xi(1,5) - xi(2,1)*xi(1,6) + xi(1,5)*xi(1,1)\\
$

{1,3}|2\\xi(4,3)*xi(2,6) + xi(3,3)*xi(2,5) - xi(2,6)*xi(2,1) + xi(2,5)*xi(1,1)\\
$

{1,3}|3\\xi(4,3)*xi(3,6) - xi(3,6)*xi(2,1) + xi(3,5)*xi(3,3) + xi(3,5)*xi(1,1)\\
$

{1,3}|4\\xi(4,6)*xi(4,3) - xi(4,6)*xi(2,1) + xi(4,5)*xi(3,3) + xi(4,5)*xi(1,1)\\
$

{1,3}|5\\xi(5,6)*xi(4,3) - xi(5,6)*xi(2,1) + xi(5,5)*xi(3,3) + xi(5,5)*xi(1,1) -
 xi(4,3)*xi(2,1) + xi(4,1)*xi(2,3) - xi(3,3)*xi(1,1) + xi(3,1)*xi(1,3) + 1\\$

{1,3}|6\\xi(6,6)*xi(4,3) - xi(6,6)*xi(2,1) + xi(6,5)*xi(3,3) + xi(6,5)*xi(1,1) -
 xi(4,3)*xi(1,1) + xi(4,1)*xi(1,3) + xi(3,3)*xi(2,1) - xi(3,1)*xi(2,3)\\$

{1,4}|1\\xi(4,4)*xi(1,6) + xi(3,4)*xi(1,5) + xi(2,1)*xi(1,5) + xi(1,6)*xi(1,1)\\
$

{1,4}|2\\xi(4,4)*xi(2,6) + xi(3,4)*xi(2,5) + xi(2,6)*xi(1,1) + xi(2,5)*xi(2,1)\\
$

{1,4}|3\\xi(4,4)*xi(3,6) + xi(3,6)*xi(1,1) + xi(3,5)*xi(3,4) + xi(3,5)*xi(2,1)\\
$

{1,4}|4\\xi(4,6)*xi(4,4) + xi(4,6)*xi(1,1) + xi(4,5)*xi(3,4) + xi(4,5)*xi(2,1)\\
$

{1,4}|5\\xi(5,6)*xi(4,4) + xi(5,6)*xi(1,1) + xi(5,5)*xi(3,4) + xi(5,5)*xi(2,1) -
 xi(4,4)*xi(2,1) + xi(4,1)*xi(2,4) - xi(3,4)*xi(1,1) + xi(3,1)*xi(1,4)\\$

{1,4}|6\\xi(6,6)*xi(4,4) + xi(6,6)*xi(1,1) + xi(6,5)*xi(3,4) + xi(6,5)*xi(2,1) -
 xi(4,4)*xi(1,1) + xi(4,1)*xi(1,4) + xi(3,4)*xi(2,1) - xi(3,1)*xi(2,4) + 1\\$

{1,5}|1\\xi(4,5)*xi(1,6) + xi(3,5)*xi(1,5)\\$

{1,5}|2\\xi(4,5)*xi(2,6) + xi(3,5)*xi(2,5)\\$

{1,5}|3\\xi(4,5)*xi(3,6) + xi(3,5)**2\\$

{1,5}|4\\xi(4,6)*xi(4,5) + xi(4,5)*xi(3,5)\\$

{1,5}|5\\xi(5,6)*xi(4,5) + xi(5,5)*xi(3,5) - xi(4,5)*xi(2,1) + xi(4,1)*xi(2,5) -
 xi(3,5)*xi(1,1) + xi(3,1)*xi(1,5)\\$

{1,5}|6\\xi(6,6)*xi(4,5) + xi(6,5)*xi(3,5) - xi(4,5)*xi(1,1) + xi(4,1)*xi(1,5) +
 xi(3,5)*xi(2,1) - xi(3,1)*xi(2,5)\\$

{1,6}|1\\xi(4,6)*xi(1,6) + xi(3,6)*xi(1,5)\\$

{1,6}|2\\xi(4,6)*xi(2,6) + xi(3,6)*xi(2,5)\\$

{1,6}|3\\xi(4,6)*xi(3,6) + xi(3,6)*xi(3,5)\\$

{1,6}|4\\xi(4,6)**2 + xi(4,5)*xi(3,6)\\$

{1,6}|5\\xi(5,6)*xi(4,6) + xi(5,5)*xi(3,6) - xi(4,6)*xi(2,1) + xi(4,1)*xi(2,6) -
 xi(3,6)*xi(1,1) + xi(3,1)*xi(1,6)\\$

{1,6}|6\\xi(6,6)*xi(4,6) + xi(6,5)*xi(3,6) - xi(4,6)*xi(1,1) + xi(4,1)*xi(1,6) +
 xi(3,6)*xi(2,1) - xi(3,1)*xi(2,6)\\$

{2,3}|1\\xi(4,3)*xi(1,5) - xi(3,3)*xi(1,6) - xi(2,2)*xi(1,6) + xi(1,5)*xi(1,2)\\
$

{2,3}|2\\xi(4,3)*xi(2,5) - xi(3,3)*xi(2,6) - xi(2,6)*xi(2,2) + xi(2,5)*xi(1,2)\\
$

{2,3}|3\\xi(4,3)*xi(3,5) - xi(3,6)*xi(3,3) - xi(3,6)*xi(2,2) + xi(3,5)*xi(1,2)\\
$

{2,3}|4\\ - xi(4,6)*xi(3,3) - xi(4,6)*xi(2,2) + xi(4,5)*xi(4,3) + xi(4,5)*xi(1,2
)\\$

{2,3}|5\\ - xi(5,6)*xi(3,3) - xi(5,6)*xi(2,2) + xi(5,5)*xi(4,3) + xi(5,5)*xi(1,2
) - xi(4,3)*xi(2,2) + xi(4,2)*xi(2,3) - xi(3,3)*xi(1,2) + xi(3,2)*xi(1,3)\\$

{2,3}|6\\ - xi(6,6)*xi(3,3) - xi(6,6)*xi(2,2) + xi(6,5)*xi(4,3) + xi(6,5)*xi(1,2
) - xi(4,3)*xi(1,2) + xi(4,2)*xi(1,3) + xi(3,3)*xi(2,2) - xi(3,2)*xi(2,3) - 1\\$

{2,4}|1\\xi(4,4)*xi(1,5) - xi(3,4)*xi(1,6) + xi(2,2)*xi(1,5) + xi(1,6)*xi(1,2)\\
$

{2,4}|2\\xi(4,4)*xi(2,5) - xi(3,4)*xi(2,6) + xi(2,6)*xi(1,2) + xi(2,5)*xi(2,2)\\
$

{2,4}|3\\xi(4,4)*xi(3,5) - xi(3,6)*xi(3,4) + xi(3,6)*xi(1,2) + xi(3,5)*xi(2,2)\\
$

{2,4}|4\\ - xi(4,6)*xi(3,4) + xi(4,6)*xi(1,2) + xi(4,5)*xi(4,4) + xi(4,5)*xi(2,2
)\\$

{2,4}|5\\ - xi(5,6)*xi(3,4) + xi(5,6)*xi(1,2) + xi(5,5)*xi(4,4) + xi(5,5)*xi(2,2
) - xi(4,4)*xi(2,2) + xi(4,2)*xi(2,4) - xi(3,4)*xi(1,2) + xi(3,2)*xi(1,4) + 1\\$

{2,4}|6\\ - xi(6,6)*xi(3,4) + xi(6,6)*xi(1,2) + xi(6,5)*xi(4,4) + xi(6,5)*xi(2,2
) - xi(4,4)*xi(1,2) + xi(4,2)*xi(1,4) + xi(3,4)*xi(2,2) - xi(3,2)*xi(2,4)\\$

{2,5}|1\\xi(4,5)*xi(1,5) - xi(3,5)*xi(1,6)\\$

{2,5}|2\\xi(4,5)*xi(2,5) - xi(3,5)*xi(2,6)\\$

{2,5}|3\\xi(4,5)*xi(3,5) - xi(3,6)*xi(3,5)\\$

{2,5}|4\\ - xi(4,6)*xi(3,5) + xi(4,5)**2\\$

{2,5}|5\\ - xi(5,6)*xi(3,5) + xi(5,5)*xi(4,5) - xi(4,5)*xi(2,2) + xi(4,2)*xi(2,5
) - xi(3,5)*xi(1,2) + xi(3,2)*xi(1,5)\\$

{2,5}|6\\ - xi(6,6)*xi(3,5) + xi(6,5)*xi(4,5) - xi(4,5)*xi(1,2) + xi(4,2)*xi(1,5
) + xi(3,5)*xi(2,2) - xi(3,2)*xi(2,5)\\$

{2,6}|1\\xi(4,6)*xi(1,5) - xi(3,6)*xi(1,6)\\$

{2,6}|2\\xi(4,6)*xi(2,5) - xi(3,6)*xi(2,6)\\$

{2,6}|3\\xi(4,6)*xi(3,5) - xi(3,6)**2\\$

{2,6}|4\\xi(4,6)*xi(4,5) - xi(4,6)*xi(3,6)\\$

{2,6}|5\\ - xi(5,6)*xi(3,6) + xi(5,5)*xi(4,6) - xi(4,6)*xi(2,2) + xi(4,2)*xi(2,6
) - xi(3,6)*xi(1,2) + xi(3,2)*xi(1,6)\\$

{2,6}|6\\ - xi(6,6)*xi(3,6) + xi(6,5)*xi(4,6) - xi(4,6)*xi(1,2) + xi(4,2)*xi(1,6
) + xi(3,6)*xi(2,2) - xi(3,2)*xi(2,6)\\$

{3,4}|1\\xi(2,4)*xi(1,6) + xi(2,3)*xi(1,5) + xi(1,6)*xi(1,3) - xi(1,5)*xi(1,4)\\
$

{3,4}|2\\xi(2,6)*xi(2,4) + xi(2,6)*xi(1,3) + xi(2,5)*xi(2,3) - xi(2,5)*xi(1,4)\\
$

{3,4}|3\\xi(3,6)*xi(2,4) + xi(3,6)*xi(1,3) + xi(3,5)*xi(2,3) - xi(3,5)*xi(1,4)\\
$

{3,4}|4\\xi(4,6)*xi(2,4) + xi(4,6)*xi(1,3) + xi(4,5)*xi(2,3) - xi(4,5)*xi(1,4)\\
$

{3,4}|5\\xi(5,6)*xi(2,4) + xi(5,6)*xi(1,3) + xi(5,5)*xi(2,3) - xi(5,5)*xi(1,4) -
 xi(4,4)*xi(2,3) + xi(4,3)*xi(2,4) - xi(3,4)*xi(1,3) + xi(3,3)*xi(1,4)\\$

{3,4}|6\\xi(6,6)*xi(2,4) + xi(6,6)*xi(1,3) + xi(6,5)*xi(2,3) - xi(6,5)*xi(1,4) -
 xi(4,4)*xi(1,3) + xi(4,3)*xi(1,4) + xi(3,4)*xi(2,3) - xi(3,3)*xi(2,4)\\$

{3,5}|1\\xi(2,5)*xi(1,6) - xi(1,5)**2\\$

{3,5}|2\\xi(2,6)*xi(2,5) - xi(2,5)*xi(1,5)\\$

{3,5}|3\\xi(3,6)*xi(2,5) - xi(3,5)*xi(1,5)\\$

{3,5}|4\\xi(4,6)*xi(2,5) - xi(4,5)*xi(1,5)\\$

{3,5}|5\\xi(5,6)*xi(2,5) - xi(5,5)*xi(1,5) - xi(4,5)*xi(2,3) + xi(4,3)*xi(2,5) -
 xi(3,5)*xi(1,3) + xi(3,3)*xi(1,5)\\$

{3,5}|6\\xi(6,6)*xi(2,5) - xi(6,5)*xi(1,5) - xi(4,5)*xi(1,3) + xi(4,3)*xi(1,5) +
 xi(3,5)*xi(2,3) - xi(3,3)*xi(2,5)\\$

{3,6}|1\\xi(2,6)*xi(1,6) - xi(1,6)*xi(1,5)\\$

{3,6}|2\\xi(2,6)**2 - xi(2,5)*xi(1,6)\\$

{3,6}|3\\xi(3,6)*xi(2,6) - xi(3,5)*xi(1,6)\\$

{3,6}|4\\xi(4,6)*xi(2,6) - xi(4,5)*xi(1,6)\\$

{3,6}|5\\xi(5,6)*xi(2,6) - xi(5,5)*xi(1,6) - xi(4,6)*xi(2,3) + xi(4,3)*xi(2,6) -
 xi(3,6)*xi(1,3) + xi(3,3)*xi(1,6)\\$

{3,6}|6\\xi(6,6)*xi(2,6) - xi(6,5)*xi(1,6) - xi(4,6)*xi(1,3) + xi(4,3)*xi(1,6) +
 xi(3,6)*xi(2,3) - xi(3,3)*xi(2,6)\\$

{4,5}|1\\ - xi(2,5)*xi(1,5) - xi(1,6)*xi(1,5)\\$

{4,5}|2\\ - xi(2,6)*xi(1,5) - xi(2,5)**2\\$

{4,5}|3\\ - xi(3,6)*xi(1,5) - xi(3,5)*xi(2,5)\\$

{4,5}|4\\ - xi(4,6)*xi(1,5) - xi(4,5)*xi(2,5)\\$

{4,5}|5\\ - xi(5,6)*xi(1,5) - xi(5,5)*xi(2,5) - xi(4,5)*xi(2,4) + xi(4,4)*xi(2,5
) - xi(3,5)*xi(1,4) + xi(3,4)*xi(1,5)\\$

{4,5}|6\\ - xi(6,6)*xi(1,5) - xi(6,5)*xi(2,5) - xi(4,5)*xi(1,4) + xi(4,4)*xi(1,5
) + xi(3,5)*xi(2,4) - xi(3,4)*xi(2,5)\\$

{4,6}|1\\ - xi(2,6)*xi(1,5) - xi(1,6)**2\\$

{4,6}|2\\ - xi(2,6)*xi(2,5) - xi(2,6)*xi(1,6)\\$

{4,6}|3\\ - xi(3,6)*xi(1,6) - xi(3,5)*xi(2,6)\\$

{4,6}|4\\ - xi(4,6)*xi(1,6) - xi(4,5)*xi(2,6)\\$

{4,6}|5\\ - xi(5,6)*xi(1,6) - xi(5,5)*xi(2,6) - xi(4,6)*xi(2,4) + xi(4,4)*xi(2,6
) - xi(3,6)*xi(1,4) + xi(3,4)*xi(1,6)\\$

{4,6}|6\\ - xi(6,6)*xi(1,6) - xi(6,5)*xi(2,6) - xi(4,6)*xi(1,4) + xi(4,4)*xi(1,6
) + xi(3,6)*xi(2,4) - xi(3,4)*xi(2,6)\\$

{5,6}|5\\ - xi(4,6)*xi(2,5) + xi(4,5)*xi(2,6) - xi(3,6)*xi(1,5) + xi(3,5)*xi(1,6
)\\$

{5,6}|6\\ - xi(4,6)*xi(1,5) + xi(4,5)*xi(1,6) + xi(3,6)*xi(2,5) - xi(3,5)*xi(2,6
)\\$

USD$

\par Simultaneous resolution of the nonzero torsion equations and the matrix$

equation USD J^2 = -I . USD$

Suppose first USDxi(1,5) \neq 0USD.$

\\ Then one first gets$

\\ from equation USD45|1USD :$

USDxi(1,6):=-xi(2,5)USD$

\\ and then from equation USD36|2USD : USD xi(2,6)**2 = -xi(2,5)**2USD, hence$

USDxi(2,6):=0,xi(2,5):=0,xi(1,6):=0USD$

But then from equation USD35|1USD : USD xi(1,5)=0USD  a contradiction$

Hence USD xi(1,5)USD has to be USD 0 USD$

\\ USD xi(1,5):=0USD$

\par With these values, \textit{Reduce} computes again all equations.$

then, one gets from  equation USD45|2USD :$

\\ USD xi(2,5):=0USD$

\\ and from equation USD46|1USD :$

\\ USD xi(1,6):=0USD$

\par With these values, \textit{Reduce} computes again all equations.$

then, one gets from  equation USD36|2USD :$

\\ USD xi(2,6):=0USD$

Suppose now USDxi(4,6) \neq 0USD.$

\\ Then one first gets$

\\ from equation USD26|4USD : which reads USD (xi(4,5)-xi(3,6))xi(4,6)=0USD$

USDxi(4,5):=xi(3,6)USD$

\\ and then from equation USD16|4USD$

which reads USD xi(4,6)$**2 + xi(4,5)xi(3,6)=0USD$

: USD xi(4,6)**2 = -xi(3,6)**2USD, hence$

USDxi(4,6):=0,xi(3,6):=0USD$

  a contradiction$

Hence USD xi(4,6)USD has to be USD 0 USD$

\\ USD xi(4,6):=0USD$

\par With these values, \textit{Reduce} computes again all equations.$

then, one gets from  equation USD25|4USD :$

\\ USD xi(4,5):=0USD$

\\ and from equation USD26|6USD :$

\\ USD xi(3,6):=0USD$

\par With these values, \textit{Reduce} computes again all equations.$

then, one gets from  equation USD15|3USD :$

\\ USD xi(3,5):=0USD$

\par With these values, \textit{Reduce} computes again all equations.$

Then the 5x5 entry in  USD J**2 USD  is $

USD  {J**2}^5_5=xi(5,5)**2 + xi(6,5)*xi(5,6);USD\\$

Hence USD xi(5,6)xi(6,5)\neq 0 USD$

and USD xi(5,6) =(-1-xi(5,5)**2)/xi(6,5). USD$

\\ USD xi(5,6):= - (xi(5,5)**2 + 1)/xi(6,5)USD$

Moreover, the 5x6 entry in  USD J**2 USD  is $

USD  (xi(5,5) + xi(6,6))*xi(5,6);USD hence\\$

\\ USD xi(6,6):= - xi(5,5)USD$

\par With these values, \textit{Reduce} computes again all equations.$

Then the 6x1 entry in  USD J**2 USD  gives $

\\ USD xi(5,1):=( - xi(6,4)*xi(4,1) - xi(6,3)*xi(3,1) - xi(6,2)*xi(2,1) + xi(6,1
)*xi(5,5) - xi(6,1)*xi(1,1))/xi(6,5)USD$

and the 6x2 entry in  USD J**2 USD  gives $

\\ USD xi(5,2):=( - xi(6,4)*xi(4,2) - xi(6,3)*xi(3,2) + xi(6,2)*xi(5,5) - xi(6,2
)*xi(2,2) - xi(6,1)*xi(1,2))/xi(6,5)USD$

and the 6x3 entry in  USD J**2 USD  gives $

\\ USD xi(5,3):=( - xi(6,4)*xi(4,3) + xi(6,3)*xi(5,5) - xi(6,3)*xi(3,3) - xi(6,2
)*xi(2,3) - xi(6,1)*xi(1,3))/xi(6,5)USD$

and the 6x4 entry in  USD J**2 USD  gives $

\\ USD xi(5,4):=(xi(6,4)*xi(5,5) - xi(6,4)*xi(4,4) - xi(6,3)*xi(3,4) - xi(6,2)*
xi(2,4) - xi(6,1)*xi(1,4))/xi(6,5)USD$

From the trace of USD JUSD, we get:$

\\ USD xi(4,4):= - (xi(3,3) + xi(2,2) + xi(1,1))USD$

\\ \textbf{Case} USDxi(2,1) \neq 0 \\USD$

\par With these values, \textit{Reduce} computes again all equations.$

Then the 1x1 entry in  USD J**2 USD  gives $

\\ USD xi(1,2):= - (xi(4,1)*xi(1,4) + xi(3,1)*xi(1,3) + xi(1,1)**2 + 1)/xi(2,1)
USD$

and the 2x1 entry in  USD J**2 USD  gives $

\\ USD xi(1,1):= - (xi(4,1)*xi(2,4) + xi(3,1)*xi(2,3) + xi(2,2)*xi(2,1))/xi(2,1)
USD$

and the 2x3 entry in  USD J**2 USD  gives $

\\ USD xi(1,3):= - (xi(4,3)*xi(2,4) + xi(3,3)*xi(2,3) + xi(2,3)*xi(2,2))/xi(2,1)
USD$

and the 2x4 entry in  USD J**2 USD  gives $

\\ USD xi(1,4):=( - xi(4,1)*xi(2,4)**2 - xi(3,4)*xi(2,3)*xi(2,1) + xi(3,3)*xi(2,
4)*xi(2,1) - xi(3,1)*xi(2,4)*xi(2,3) - xi(2,4)*xi(2,2)*xi(2,1))/xi(2,1)**2USD$

\par With these values, \textit{Reduce} computes again all equations.$

Then the 4x1 entry in  USD J**2 USD  gives $

\\ USD xi(4,2):=( - xi(4,3)*xi(3,1) + xi(4,1)*xi(3,3) + xi(4,1)*xi(2,2))/xi(2,1)
USD$

\par With these values, \textit{Reduce} computes again all equations.$

Then the 3x1 entry in  USD J**2 USD  gives $

\\ USD xi(3,2):=( - xi(4,1)*xi(3,4)*xi(2,1) + xi(4,1)*xi(3,1)*xi(2,4) - xi(3,3)*
xi(3,1)*xi(2,1) + xi(3,1)**2*xi(2,3) + xi(3,1)*xi(2,2)*xi(2,1))/xi(2,1)**2USD\\$

The 4x4 entry in  USD J**2 USD  reads $

USD(xi(4,3)*xi(2,1)- xi(4,1)*xi(2,3))(xi(3,4)*xi(2,1) -xi(3,1)*xi(2,4))$

+(xi(3,3)*xi(2,1) - xi(3,1)*xi(2,3))**2)/xi(2,1)**2 =-1USD$

Hence the case USD(xi(4,3)*xi(2,1)- xi(4,1)*xi(2,3)) =0USD is impossible$

and the case USD(xi(3,4)*xi(2,1)- xi(3,1)*xi(2,4)) =0USD is impossible as well.$

We hence have the 2 conditions$

condition1 : USD C1:=xi^4_3 xi^2_1 - xi^4_1 xi^2_3 \neq 0USD$

condition2 : USD C2:=xi^3_4 xi^2_1 - xi^3_1 xi^2_4 \neq 0USD$

Then we get :$

\\ USD xi(4,3):=(xi(4,1)*xi(3,4)*xi(2,3)*xi(2,1) - xi(4,1)*xi(3,1)*xi(2,4)*xi(2,
3) - xi(3,3)**2*xi(2,1)**2 + 2*xi(3,3)*xi(3,1)*xi(2,3)*xi(2,1) - xi(3,1)**2*xi(2
,3)**2 - xi(2,1)**2)/(xi(2,1)*(xi(3,4)*xi(2,1) - xi(3,1)*xi(2,4)))USD$

\\ \textbf{Case} USDxi(2,3) = 0 \\USD$

\par Suppose USD xi(2,3) =    0 USD.$

\\ USD xi(2,3):=0USD$

\\ \textbf{Case} USDxi(2,4) = 0 \\USD$

\par Suppose USD xi(2,4) =    0 USD.$

\\ USD xi(2,4):=0USD$

then, one gets from  equation USD14|6USD :$

\par With these values, \textit{Reduce} computes again all equations.$

condition1 reads then : USD  - (xi(2,1)*(xi(3,3)**2 + 1))/xi(3,4)\neq 0 USD$

and condition2 reads : USD xi(3,4)*xi(2,1)\neq 0 .USD$

\\ \textbf{Case} USDxi(3,3):=- xi(2,2) \ \\USD$

\par Suppose USD xi(3,3):=-xi(2,2) USD.$

\\ USD xi(3,3):= - xi(2,2)USD$

\par With these values, \textit{Reduce} computes again all equations.$

\\ \textbf{Case} USDxi(3,4) \neq -xi(2,1)  \\USD$

\par Suppose       USD xi(3,4) \neq -xi(2,1)  USD.$

Then,   equation USD14|5USD gives :$

\\ USD xi(5,5):=(xi(2,2)*( - xi(3,4) + xi(2,1)))/(xi(3,4) + xi(2,1))USD$

and,   equation USD14|6USD gives :$

\\ USD xi(6,5):= - (xi(3,4)*xi(2,1) + xi(2,2)**2 + 1)/(xi(3,4) + xi(2,1))USD$

 \par Now the nonzero torsion equations left are :$

Torsion equations to cancel (Latex output) : USD$

USD$

\\ \P \\$

\par The matrix USD J USD is :\\$

with the condition USD C1:=( - (xi(2,2)**2 + 1)*xi(2,1))/xi(3,4)\neq 0;USD$

and the condition  USD C2:=xi(3,4)*xi(2,1)\neq 0;USD$

and  USDxi(3,4)\neq - xi(2,1) \ \\USD$

and USD xi(6,5):= - (xi(2,2)**2 + 1 + xi(3,4)*xi(2,1))/(xi(3,4) + xi(2,1))
 \neq 0  \\USD$

USD  J^1_1= - xi(2,2);USD\\$

USD  J^1_2= - (xi(2,2)**2 + 1)/xi(2,1);USD\\$

USD  J^1_3=0;USD\\$

USD  J^1_4=0;USD\\$

USD  J^1_5=0;USD\\$

USD  J^1_6=0;USD\\$

USD  J^2_1=xi(2,1);USD\\$

USD  J^2_2=xi(2,2);USD\\$

USD  J^2_3=0;USD\\$

USD  J^2_4=0;USD\\$

USD  J^2_5=0;USD\\$

USD  J^2_6=0;USD\\$

USD  J^3_1=xi(3,1);USD\\$

USD  J^3_2=( - xi(4,1)*xi(3,4) + 2*xi(3,1)*xi(2,2))/xi(2,1);USD\\$

USD  J^3_3= - xi(2,2);USD\\$

USD  J^3_4=xi(3,4);USD\\$

USD  J^3_5=0;USD\\$

USD  J^3_6=0;USD\\$

USD  J^4_1=xi(4,1);USD\\$

USD  J^4_2=(xi(3,1)*(xi(2,2)**2 + 1))/(xi(3,4)*xi(2,1));USD\\$

USD  J^4_3= - (xi(2,2)**2 + 1)/xi(3,4);USD\\$

USD  J^4_4=xi(2,2);USD\\$

USD  J^4_5=0;USD\\$

USD  J^4_6=0;USD\\$

USD  J^5_1=(xi(6,4)*xi(4,1)*xi(3,4) + xi(6,4)*xi(4,1)*xi(2,1) + xi(6,3)*xi(3,4)*
xi(3,1) + xi(6,3)*xi(3,1)*xi(2,1) + xi(6,2)*xi(3,4)*xi(2,1) + xi(6,2)*xi(2,1)**2
 - 2*xi(6,1)*xi(2,2)*xi(2,1))/(xi(3,4)*xi(2,1) + xi(2,2)**2 + 1);USD\\$

USD  J^5_2=(xi(6,4)*xi(3,4)*xi(3,1)*xi(2,2)**2 + xi(6,4)*xi(3,4)*xi(3,1) + xi(6,
4)*xi(3,1)*xi(2,2)**2*xi(2,1) + xi(6,4)*xi(3,1)*xi(2,1) - xi(6,3)*xi(4,1)*xi(3,4
)**3 - xi(6,3)*xi(4,1)*xi(3,4)**2*xi(2,1) + 2*xi(6,3)*xi(3,4)**2*xi(3,1)*xi(2,2)
 + 2*xi(6,3)*xi(3,4)*xi(3,1)*xi(2,2)*xi(2,1) + 2*xi(6,2)*xi(3,4)**2*xi(2,2)*xi(2
,1) - xi(6,1)*xi(3,4)**2*xi(2,2)**2 - xi(6,1)*xi(3,4)**2 - xi(6,1)*xi(3,4)*xi(2,
2)**2*xi(2,1) - xi(6,1)*xi(3,4)*xi(2,1))/(xi(3,4)*xi(2,1)*(xi(3,4)*xi(2,1) + xi(
2,2)**2 + 1));USD\\$

USD  J^5_3=( - xi(6,4)*xi(3,4)*xi(2,2)**2 - xi(6,4)*xi(3,4) - xi(6,4)*xi(2,2)**2
*xi(2,1) - xi(6,4)*xi(2,1) - 2*xi(6,3)*xi(3,4)*xi(2,2)*xi(2,1))/(xi(3,4)*(xi(3,4
)*xi(2,1) + xi(2,2)**2 + 1));USD\\$

USD  J^5_4=(xi(3,4)*(2*xi(6,4)*xi(2,2) + xi(6,3)*xi(3,4) + xi(6,3)*xi(2,1)))/(xi
(3,4)*xi(2,1) + xi(2,2)**2 + 1);USD\\$

USD  J^5_5=(xi(2,2)*( - xi(3,4) + xi(2,1)))/(xi(3,4) + xi(2,1));USD\\$

USD  J^5_6=(xi(3,4)**2*xi(2,2)**2 + xi(3,4)**2 - 2*xi(3,4)*xi(2,2)**2*xi(2,1) + 
2*xi(3,4)*xi(2,1) + xi(2,2)**2*xi(2,1)**2 + xi(2,1)**2)/(xi(3,4)**2*xi(2,1) + xi
(3,4)*xi(2,2)**2 + xi(3,4)*xi(2,1)**2 + xi(3,4) + xi(2,2)**2*xi(2,1) + xi(2,1))
;USD\\$

USD  J^6_1=xi(6,1);USD\\$

USD  J^6_2=xi(6,2);USD\\$

USD  J^6_3=xi(6,3);USD\\$

USD  J^6_4=xi(6,4);USD\\$

USD  J^6_5= - (xi(3,4)*xi(2,1) + xi(2,2)**2 + 1)/(xi(3,4) + xi(2,1));USD\\$

USD  J^6_6=(xi(2,2)*(xi(3,4) - xi(2,1)))/(xi(3,4) + xi(2,1));USD\\$

USDUSD J^2 = \begin{pmatrix}$

-1&$

0&$

0&$

0&$

0&$

0\\$

0&$

-1&$

0&$

0&$

0&$

0\\$

0&$

0&$

-1&$

0&$

0&$

0\\$

0&$

0&$

0&$

-1&$

0&$

0\\$

0&$

0&$

0&$

0&$

-1&$

0\\$

0&$

0&$

0&$

0&$

0&$

-1\end{pmatrix}USDUSD$

\\$ det J:=1$

Trace J:=0$

\end{document}$