\documentclass{article} \usepackage{amsmath,amssymb} \sloppy \begin{document} Computation of all complex structures on the real Lie Algebra USD {\mathcal{G}}_{6,n2}.USD \\Case 3. \smallskip \par Commutation relations for USD {\mathcal{G}}_{6,n2}:USD\\ USD[x(1),x(2)]=x(5)USD; USD[x(3),x(4)]=x(6)USD; \P Nonzero torsion \par Torsion equations to cancel (Latex output) : \\USD {1,2}|1\\xi(2,2)*xi(1,5) + xi(1,5)*xi(1,1)\\ {1,2}|2\\xi(2,5)*xi(2,2) + xi(2,5)*xi(1,1)\\ {1,2}|3\\xi(3,5)*xi(2,2) + xi(3,5)*xi(1,1)\\ {1,2}|4\\xi(4,5)*xi(2,2) + xi(4,5)*xi(1,1)\\ {1,2}|5\\xi(2,1)*xi(1,2) + 1 - xi(2,2)*xi(1,1) + (xi(2,2) + xi(1,1))*xi(5,5)\\ {1,2}|6\\ - (xi(4,2)*xi(3,1) - xi(4,1)*xi(3,2)) + (xi(2,2) + xi(1,1))*xi(6,5)\\ {1,3}|1\\ - xi(4,1)*xi(1,6) + xi(2,3)*xi(1,5)\\ {1,3}|2\\ - xi(4,1)*xi(2,6) + xi(2,5)*xi(2,3)\\ {1,3}|3\\ - xi(4,1)*xi(3,6) + xi(3,5)*xi(2,3)\\ {1,3}|4\\ - xi(4,6)*xi(4,1) + xi(4,5)*xi(2,3)\\ {1,3}|5\\ - (xi(2,3)*xi(1,1) - xi(2,1)*xi(1,3) - xi(5,5)*xi(2,3)) - xi(5,6)*xi(4 ,1)\\ {1,3}|6\\ - (xi(4,3)*xi(3,1) - xi(4,1)*xi(3,3) - xi(6,5)*xi(2,3)) - xi(6,6)*xi(4 ,1)\\ {1,4}|1\\xi(3,1)*xi(1,6) + xi(2,4)*xi(1,5)\\ {1,4}|2\\xi(3,1)*xi(2,6) + xi(2,5)*xi(2,4)\\ {1,4}|3\\xi(3,6)*xi(3,1) + xi(3,5)*xi(2,4)\\ {1,4}|4\\xi(4,6)*xi(3,1) + xi(4,5)*xi(2,4)\\ {1,4}|5\\ - (xi(2,4)*xi(1,1) - xi(2,1)*xi(1,4) - xi(5,5)*xi(2,4)) + xi(5,6)*xi(3 ,1)\\ {1,4}|6\\ - (xi(4,4)*xi(3,1) - xi(4,1)*xi(3,4) - xi(6,5)*xi(2,4)) + xi(6,6)*xi(3 ,1)\\ {1,5}|1\\xi(2,5)*xi(1,5)\\ {1,5}|2\\xi(2,5)**2\\ {1,5}|3\\xi(3,5)*xi(2,5)\\ {1,5}|4\\xi(4,5)*xi(2,5)\\ {1,5}|5\\ - (xi(2,5)*xi(1,1) - xi(2,1)*xi(1,5)) + xi(5,5)*xi(2,5)\\ {1,5}|6\\ - (xi(4,5)*xi(3,1) - xi(4,1)*xi(3,5)) + xi(6,5)*xi(2,5)\\ {1,6}|1\\xi(2,6)*xi(1,5)\\ {1,6}|2\\xi(2,6)*xi(2,5)\\ {1,6}|3\\xi(3,5)*xi(2,6)\\ {1,6}|4\\xi(4,5)*xi(2,6)\\ {1,6}|5\\ - (xi(2,6)*xi(1,1) - xi(2,1)*xi(1,6)) + xi(5,5)*xi(2,6)\\ {1,6}|6\\ - (xi(4,6)*xi(3,1) - xi(4,1)*xi(3,6)) + xi(6,5)*xi(2,6)\\ {2,3}|1\\ - xi(4,2)*xi(1,6) - xi(1,5)*xi(1,3)\\ {2,3}|2\\ - xi(4,2)*xi(2,6) - xi(2,5)*xi(1,3)\\ {2,3}|3\\ - xi(4,2)*xi(3,6) - xi(3,5)*xi(1,3)\\ {2,3}|4\\ - xi(4,6)*xi(4,2) - xi(4,5)*xi(1,3)\\ {2,3}|5\\ - (xi(2,3)*xi(1,2) - xi(2,2)*xi(1,3) + xi(5,5)*xi(1,3)) - xi(5,6)*xi(4 ,2)\\ {2,3}|6\\ - (xi(4,3)*xi(3,2) - xi(4,2)*xi(3,3) + xi(6,5)*xi(1,3)) - xi(6,6)*xi(4 ,2)\\ {2,4}|1\\xi(3,2)*xi(1,6) - xi(1,5)*xi(1,4)\\ {2,4}|2\\xi(3,2)*xi(2,6) - xi(2,5)*xi(1,4)\\ {2,4}|3\\xi(3,6)*xi(3,2) - xi(3,5)*xi(1,4)\\ {2,4}|4\\xi(4,6)*xi(3,2) - xi(4,5)*xi(1,4)\\ {2,4}|5\\ - (xi(2,4)*xi(1,2) - xi(2,2)*xi(1,4) + xi(5,5)*xi(1,4)) + xi(5,6)*xi(3 ,2)\\ {2,4}|6\\ - (xi(4,4)*xi(3,2) - xi(4,2)*xi(3,4) + xi(6,5)*xi(1,4)) + xi(6,6)*xi(3 ,2)\\ {2,5}|1\\ - xi(1,5)**2\\ {2,5}|2\\ - xi(2,5)*xi(1,5)\\ {2,5}|3\\ - xi(3,5)*xi(1,5)\\ {2,5}|4\\ - xi(4,5)*xi(1,5)\\ {2,5}|5\\ - (xi(2,5)*xi(1,2) - xi(2,2)*xi(1,5)) - xi(5,5)*xi(1,5)\\ {2,5}|6\\ - (xi(4,5)*xi(3,2) - xi(4,2)*xi(3,5)) - xi(6,5)*xi(1,5)\\ {2,6}|1\\ - xi(1,6)*xi(1,5)\\ {2,6}|2\\ - xi(2,5)*xi(1,6)\\ {2,6}|3\\ - xi(3,5)*xi(1,6)\\ {2,6}|4\\ - xi(4,5)*xi(1,6)\\ {2,6}|5\\ - (xi(2,6)*xi(1,2) - xi(2,2)*xi(1,6)) - xi(5,5)*xi(1,6)\\ {2,6}|6\\ - (xi(4,6)*xi(3,2) - xi(4,2)*xi(3,6)) - xi(6,5)*xi(1,6)\\ {3,4}|1\\xi(4,4)*xi(1,6) + xi(3,3)*xi(1,6)\\ {3,4}|2\\xi(4,4)*xi(2,6) + xi(3,3)*xi(2,6)\\ {3,4}|3\\xi(4,4)*xi(3,6) + xi(3,6)*xi(3,3)\\ {3,4}|4\\xi(4,6)*xi(4,4) + xi(4,6)*xi(3,3)\\ {3,4}|5\\(xi(4,4) + xi(3,3))*xi(5,6) - (xi(2,4)*xi(1,3) - xi(2,3)*xi(1,4))\\ {3,4}|6\\xi(4,3)*xi(3,4) + 1 - xi(4,4)*xi(3,3) + (xi(4,4) + xi(3,3))*xi(6,6)\\ {3,5}|1\\xi(4,5)*xi(1,6)\\ {3,5}|2\\xi(4,5)*xi(2,6)\\ {3,5}|3\\xi(4,5)*xi(3,6)\\ {3,5}|4\\xi(4,6)*xi(4,5)\\ {3,5}|5\\ - (xi(2,5)*xi(1,3) - xi(2,3)*xi(1,5)) + xi(5,6)*xi(4,5)\\ {3,5}|6\\ - (xi(4,5)*xi(3,3) - xi(4,3)*xi(3,5)) + xi(6,6)*xi(4,5)\\ {3,6}|1\\xi(4,6)*xi(1,6)\\ {3,6}|2\\xi(4,6)*xi(2,6)\\ {3,6}|3\\xi(4,6)*xi(3,6)\\ {3,6}|4\\xi(4,6)**2\\ {3,6}|5\\ - (xi(2,6)*xi(1,3) - xi(2,3)*xi(1,6)) + xi(5,6)*xi(4,6)\\ {3,6}|6\\ - (xi(4,6)*xi(3,3) - xi(4,3)*xi(3,6)) + xi(6,6)*xi(4,6)\\ {4,5}|1\\ - xi(3,5)*xi(1,6)\\ {4,5}|2\\ - xi(3,5)*xi(2,6)\\ {4,5}|3\\ - xi(3,6)*xi(3,5)\\ {4,5}|4\\ - xi(4,6)*xi(3,5)\\ {4,5}|5\\ - (xi(2,5)*xi(1,4) - xi(2,4)*xi(1,5)) - xi(5,6)*xi(3,5)\\ {4,5}|6\\ - (xi(4,5)*xi(3,4) - xi(4,4)*xi(3,5)) - xi(6,6)*xi(3,5)\\ {4,6}|1\\ - xi(3,6)*xi(1,6)\\ {4,6}|2\\ - xi(3,6)*xi(2,6)\\ {4,6}|3\\ - xi(3,6)**2\\ {4,6}|4\\ - xi(4,6)*xi(3,6)\\ {4,6}|5\\ - (xi(2,6)*xi(1,4) - xi(2,4)*xi(1,6)) - xi(5,6)*xi(3,6)\\ {4,6}|6\\ - (xi(4,6)*xi(3,4) - xi(4,4)*xi(3,6)) - xi(6,6)*xi(3,6)\\ {5,6}|5\\ - xi(2,6)*xi(1,5) + xi(2,5)*xi(1,6)\\ {5,6}|6\\ - xi(4,6)*xi(3,5) + xi(4,5)*xi(3,6)\\ USD \par Simultaneous resolution of the nonzero torsion equations and the matrix equation USD J^2 = -I . USD \\ One first gets \\ from equation USD15|2USD : \\ USD xi(2,5):=0USD \\ and from equation USD25|1USD : \\ USD xi(1,5):=0USD \\ and from equation USD36|4USD : \\ USD xi(4,6):=0USD \\ and from equation USD46|3USD : \\ USD xi(3,6):=0USD ******************************************************************* Here in case 3, we suppose that : \\ USD xi(1,1):=0USD \\ USD xi(2,1):=1USD \\ USD xi(5,1):=0USD \\ USD xi(5,2):=0USD \\ USD xi(4,4):=0USD \\ USD xi(3,4):= - xi(3,3)USD \\ USD xi(6,4):=0USD \\ USD xi(6,3):=0USD ******************************************************************* Then from equation %{{{1,2},3},xi(3,5)*xi(2,2)}, \\ USD xi(3,5):=0USD %{{{1,2},4},xi(4,5)*xi(2,2)}, \\ USD xi(4,5):=0USD %{{{3,4},1},xi(3,3)*xi(1,6)}, \\ USD xi(1,6):=0USD %{{{3,4},2},xi(3,3)*xi(2,6)}, \\ USD xi(2,6):=0USD Then from equations. %\\USD J^2(6,5):=(xi(6,6) + xi(5,5))*xi(6,5)USD\\ %\\USD J^2(6,6):=xi(6,6)**2 + xi(6,5)*xi(5,6)USD\\ \\ USD xi(6,6):= - xi(5,5)USD and from the trace %Trace(J):=xi(3,3) + xi(2,2) + xi(5,5) + xi(6,6)$ \\ USD xi(2,2):= - xi(3,3)USD Then from the equations %{{{1,2},5},xi(1,2) + 1 - xi(5,5)*xi(3,3)}, %{{{3,4},6}, - (xi(4,3)*xi(3,3) - 1 + xi(5,5)*xi(3,3))}, \\ USD xi(1,2):= - xi(4,3)*xi(3,3)USD Then from the equation %\\USD J^2(6,1):=xi(6,2) - xi(6,1)*xi(5,5)USD\\$ \\ USD xi(6,2):=xi(6,1)*xi(5,5)USD Then from the equations %{{{3,4},6}, - xi(5,5)*xi(3,3) - xi(4,3)*xi(3,3) + 1}, !%!\!\!U!S!D! !J^2(6,2):=! -! xi(6,1)*(xi(5,5)**2! +! xi(5,5)*xi(3,3)! +! xi(4,3 )*xi(3,3))!U!S\ !D!\!\$! \\ USD xi(6,1):=0USD Then from the equations %\\USD J^2(1,4):= - xi(3,3)*(xi(4,3)*xi(2,4) + xi(1,3))USD\\$ \\ USD xi(1,3):= - xi(4,3)*xi(2,4)USD %\\USD J^2(4,1):=xi(4,3)*xi(3,1) + xi(4,2)USD\\$ \\ USD xi(4,2):= - xi(4,3)*xi(3,1)USD %\\USD J^2(6,3):=xi(6,5)*xi(5,3)USD\\$ %\\USD J^2(6,4):=xi(6,5)*xi(5,4)USD\\$ \\ USD xi(5,3):=0USD \\ USD xi(5,4):=0USD Then from the equations %{{{1,2},5}, - (xi(4,3)*xi(3,3) - 1 + xi(5,5)*xi(3,3))}, \\ USD xi(5,5):=( - (xi(4,3)*xi(3,3) - 1))/xi(3,3)USD %\\USD J^2(6,6):=xi(6,5)*xi(5,6) + xi(5,5)**2USD\\ \\ USD xi(5,6):=( - ((xi(4,3)*xi(3,3) - 1)**2 + xi(3,3)**2))/(xi(6,5)*xi(3,3)**2 )USD %\\USD J^2(3,1):=xi(3,3)*xi(3,1) + xi(3,2) - xi(4,1)*xi(3,3)USD\\ \\ USD xi(4,1):=(xi(3,3)*xi(3,1) + xi(3,2))/xi(3,3)USD localrecap USD \par Now the nonzero torsion equations left are : {{{{1,2},1},0}, {{{1,2},2},0}, {{{1,2},3},0}, {{{1,2},4},0}, {{{1,2},5},0}, {{{1,2},6}, ((xi(3,3)*xi(3,1) + xi(3,2))*xi(3,2) + xi(4,3)*xi(3,3)*xi(3,1)**2 - xi(6,5)*xi(3 ,3)**2)/xi(3,3)}, {{{1,3},1},0}, {{{1,3},2},0}, {{{1,3},3},0}, {{{1,3},4},0}, {{{1,3},5}, ( - (((xi(4,3)*xi(3,3)*xi(2,4) + xi(4,3)*xi(3,3)*xi(2,3) - xi(2,3))*xi(6,5) - ( xi(3,3)*xi(3,1) + xi(3,2)))*xi(3,3)**2 - (xi(4,3)*xi(3,3) - 1)**2*(xi(3,3)*xi(3, 1) + xi(3,2))))/(xi(6,5)*xi(3,3)**3)}, {{{1,3},6}, ((xi(3,3)**2 + 1)*(xi(3,3)*xi(3,1) + xi(3,2)) - (2*xi(3,3)*xi(3,1) + xi(3,2))*xi (4,3)*xi(3,3) + xi(6,5)*xi(3,3)**2*xi(2,3))/xi(3,3)**2}, {{{1,4},1},0}, {{{1,4},2},0}, {{{1,4},3},0}, {{{1,4},4},0}, {{{1,4},5}, ( - (((xi(6,5)*xi(4,3)*xi(2,4) + xi(3,1))*xi(3,3) - (xi(3,3)*xi(1,4) + xi(2,4))* xi(6,5))*xi(3,3) + (xi(4,3)*xi(3,3) - 1)**2*xi(3,1)))/(xi(6,5)*xi(3,3)**2)}, {{{1,4},6}, ( - (xi(3,3)**2*xi(3,1) + xi(3,3)*xi(3,2) + xi(3,1) - xi(4,3)*xi(3,3)*xi(3,1) - xi(6,5)*xi(3,3)*xi(2,4)))/xi(3,3)}, {{{1,5},1},0}, {{{1,5},2},0}, {{{1,5},3},0}, {{{1,5},4},0}, {{{1,5},5},0}, {{{1,5},6},0}, {{{1,6},1},0}, {{{1,6},2},0}, {{{1,6},3},0}, {{{1,6},4},0}, {{{1,6},5},0}, {{{1,6},6},0}, {{{2,3},1},0}, {{{2,3},2},0}, {{{2,3},3},0}, {{{2,3},4},0}, {{{2,3},5}, ( - ((xi(6,5)*xi(4,3)*xi(3,3)*xi(2,4) - xi(6,5)*xi(3,3)**2*xi(2,4) - xi(6,5)*xi( 3,3)**2*xi(2,3) - xi(6,5)*xi(2,4) + xi(3,3)*xi(3,1))*xi(3,3) + (xi(4,3)*xi(3,3) - 1)**2*xi(3,1))*xi(4,3))/(xi(6,5)*xi(3,3)**2)}, {{{2,3},6}, ((xi(4,3)*xi(3,3)*xi(3,1) - xi(3,3)**2*xi(3,1) - xi(3,3)*xi(3,2) - xi(3,1) + xi( 6,5)*xi(3,3)*xi(2,4))*xi(4,3))/xi(3,3)}, {{{2,4},1},0}, {{{2,4},2},0}, {{{2,4},3},0}, {{{2,4},4},0}, {{{2,4},5}, (((xi(6,5)*xi(4,3)*xi(3,3)*xi(2,4) + xi(6,5)*xi(4,3)*xi(1,4) - xi(3,2))*xi(3,3) - (xi(3,3)**2 + 1)*xi(6,5)*xi(1,4))*xi(3,3) - (xi(4,3)*xi(3,3) - 1)**2*xi(3,2))/ (xi(6,5)*xi(3,3)**2)}, {{{2,4},6}, (xi(4,3)*xi(3,3)**2*xi(3,1) + xi(4,3)*xi(3,3)*xi(3,2) - xi(3,2) - xi(6,5)*xi(3,3 )*xi(1,4))/xi(3,3)}, {{{2,5},1},0}, {{{2,5},2},0}, {{{2,5},3},0}, {{{2,5},4},0}, {{{2,5},5},0}, {{{2,5},6},0}, {{{2,6},1},0}, {{{2,6},2},0}, {{{2,6},3},0}, {{{2,6},4},0}, {{{2,6},5},0}, {{{2,6},6},0}, {{{3,4},1},0}, {{{3,4},2},0}, {{{3,4},3},0}, {{{3,4},4},0}, {{{3,4},5}, ((xi(6,5)*xi(4,3)*xi(2,4)**2 + xi(6,5)*xi(2,3)*xi(1,4) - xi(3,3))*xi(3,3) - (xi( 4,3)*xi(3,3) - 1)**2)/(xi(6,5)*xi(3,3))}, {{{3,4},6},0}, {{{3,5},1},0}, {{{3,5},2},0}, {{{3,5},3},0}, {{{3,5},4},0}, {{{3,5},5},0}, {{{3,5},6},0}, {{{3,6},1},0}, {{{3,6},2},0}, {{{3,6},3},0}, {{{3,6},4},0}, {{{3,6},5},0}, {{{3,6},6},0}, {{{4,5},1},0}, {{{4,5},2},0}, {{{4,5},3},0}, {{{4,5},4},0}, {{{4,5},5},0}, {{{4,5},6},0}, {{{4,6},1},0}, {{{4,6},2},0}, {{{4,6},3},0}, {{{4,6},4},0}, {{{4,6},5},0}, {{{4,6},6},0}, {{{5,6},5},0}, {{{5,6},6},0}} \par The matrix USD J USD is :\\ USD J^1_1=0;USD\\ USD J^1_2= - xi(4,3)*xi(3,3);USD\\ USD J^1_3= - xi(4,3)*xi(2,4);USD\\ USD J^1_4=xi(1,4);USD\\ USD J^1_5=0;USD\\ USD J^1_6=0;USD\\ USD J^2_1=1;USD\\ USD J^2_2= - xi(3,3);USD\\ USD J^2_3=xi(2,3);USD\\ USD J^2_4=xi(2,4);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(3,2);USD\\ USD J^3_3=xi(3,3);USD\\ USD J^3_4= - xi(3,3);USD\\ USD J^3_5=0;USD\\ USD J^3_6=0;USD\\ USD J^4_1=(xi(3,3)*xi(3,1) + xi(3,2))/xi(3,3);USD\\ USD J^4_2= - xi(4,3)*xi(3,1);USD\\ USD J^4_3=xi(4,3);USD\\ USD J^4_4=0;USD\\ USD J^4_5=0;USD\\ USD J^4_6=0;USD\\ USD J^5_1=0;USD\\ USD J^5_2=0;USD\\ USD J^5_3=0;USD\\ USD J^5_4=0;USD\\ USD J^5_5=( - (xi(4,3)*xi(3,3) - 1))/xi(3,3);USD\\ USD J^5_6=( - ((xi(4,3)*xi(3,3) - 1)**2 + xi(3,3)**2))/(xi(6,5)*xi(3,3)**2) ;USD\\ USD J^6_1=0;USD\\ USD J^6_2=0;USD\\ USD J^6_3=0;USD\\ USD J^6_4=0;USD\\ USD J^6_5=xi(6,5);USD\\ USD J^6_6=(xi(4,3)*xi(3,3) - 1)/xi(3,3);USD\\ \\USD J^2(1,1):=((xi(3,3)*xi(3,1) + xi(3,2))*xi(1,4) - (xi(3,3) + xi(3,1)*xi(2,4 ))*xi(4,3)*xi(3,3))/xi(3,3)USD\\ \\USD J^2(1,2):= - (xi(3,2)*xi(2,4) + xi(3,1)*xi(1,4) - xi(3,3)**2)*xi(4,3)USD\\ \\USD J^2(1,3):= - (xi(3,3)*xi(2,4) + xi(3,3)*xi(2,3) - xi(1,4))*xi(4,3)USD\\ \\USD J^2(1,4):=0USD\\ \\USD J^2(1,5):=0USD\\ \\USD J^2(1,6):=0USD\\ \\USD J^2(2,1):=( - (xi(3,3)**2 - xi(3,3)*xi(3,1)*xi(2,4) - xi(3,3)*xi(3,1)*xi(2 ,3) - xi(3,2)*xi(2,4)))/xi(3,3)USD\\ \\USD J^2(2,2):=xi(3,3)**2 + xi(3,2)*xi(2,3) - (xi(3,3) + xi(3,1)*xi(2,4))*xi(4, 3)USD\\ \\USD J^2(2,3):=0USD\\ \\USD J^2(2,4):= - (xi(3,3)*xi(2,4) + xi(3,3)*xi(2,3) - xi(1,4))USD\\ \\USD J^2(2,5):=0USD\\ \\USD J^2(2,6):=0USD\\ \\USD J^2(3,1):=0USD\\ \\USD J^2(3,2):=0USD\\ \\USD J^2(3,3):=xi(3,3)**2 + xi(3,2)*xi(2,3) - (xi(3,3) + xi(3,1)*xi(2,4))*xi(4, 3)USD\\ \\USD J^2(3,4):=xi(3,2)*xi(2,4) + xi(3,1)*xi(1,4) - xi(3,3)**2USD\\ \\USD J^2(3,5):=0USD\\ \\USD J^2(3,6):=0USD\\ \\USD J^2(4,1):=0USD\\ \\USD J^2(4,2):=0USD\\ \\USD J^2(4,3):=((xi(3,3)**2 - xi(3,3)*xi(3,1)*xi(2,4) - xi(3,3)*xi(3,1)*xi(2,3) - xi(3,2)*xi(2,4))*xi(4,3))/xi(3,3)USD\\ \\USD J^2(4,4):=((xi(3,3)*xi(3,1) + xi(3,2))*xi(1,4) - (xi(3,3) + xi(3,1)*xi(2,4 ))*xi(4,3)*xi(3,3))/xi(3,3)USD\\ \\USD J^2(4,5):=0USD\\ \\USD J^2(4,6):=0USD\\ \\USD J^2(5,1):=0USD\\ \\USD J^2(5,2):=0USD\\ \\USD J^2(5,3):=0USD\\ \\USD J^2(5,4):=0USD\\ \\USD J^2(5,5):=-1USD\\ \\USD J^2(5,6):=0USD\\ \\USD J^2(6,1):=0USD\\ \\USD J^2(6,2):=0USD\\ \\USD J^2(6,3):=0USD\\ \\USD J^2(6,4):=0USD\\ \\USD J^2(6,5):=0USD\\ \\USD J^2(6,6):=-1USD\\ Trace(J):=0 J:= mat((0, - xi(4,3)*xi(3,3), - xi(4,3)*xi(2,4),xi(1,4),0,0), (1, - xi(3,3),xi(2,3),xi(2,4),0,0), (xi(3,1),xi(3,2),xi(3,3), - xi(3,3),0,0), xi(3,3)*xi(3,1) + xi(3,2) (---------------------------, - xi(4,3)*xi(3,1),xi(4,3),0,0,0), xi(3,3) 2 2 - (xi(4,3)*xi(3,3) - 1) - ((xi(4,3)*xi(3,3) - 1) + xi(3,3) ) (0,0,0,0,--------------------------,---------------------------------------- xi(3,3) 2 xi(6,5)*xi(3,3) ), xi(4,3)*xi(3,3) - 1 (0,0,0,0,xi(6,5),---------------------)) xi(3,3) J**2:= mat((((xi(3,3)*xi(3,1) + xi(3,2))*xi(1,4) - (xi(3,3) + xi(3,1)*xi(2,4))*xi(4,3)*xi(3,3))/xi(3,3), 2 - (xi(3,2)*xi(2,4) + xi(3,1)*xi(1,4) - xi(3,3) )*xi(4,3), - (xi(3,3)*xi(2,4) + xi(3,3)*xi(2,3) - xi(1,4))*xi(4,3),0,0,0), 2 (( - (xi(3,3) - xi(3,3)*xi(3,1)*xi(2,4) - xi(3,3)*xi(3,1)*xi(2,3) - xi(3,2)*xi(2,4)))/xi(3,3), 2 xi(3,3) + xi(3,2)*xi(2,3) - (xi(3,3) + xi(3,1)*xi(2,4))*xi(4,3),0, - (xi(3,3)*xi(2,4) + xi(3,3)*xi(2,3) - xi(1,4)),0,0), 2 (0,0,xi(3,3) + xi(3,2)*xi(2,3) - (xi(3,3) + xi(3,1)*xi(2,4))*xi(4,3), 2 xi(3,2)*xi(2,4) + xi(3,1)*xi(1,4) - xi(3,3) ,0,0), 2 (0,0,((xi(3,3) - xi(3,3)*xi(3,1)*xi(2,4) - xi(3,3)*xi(3,1)*xi(2,3) - xi(3,2)*xi(2,4))*xi(4,3))/xi(3,3),( (xi(3,3)*xi(3,1) + xi(3,2))*xi(1,4) - (xi(3,3) + xi(3,1)*xi(2,4))*xi(4,3)*xi(3,3))/xi(3,3),0,0), (0,0,0,0,-1,0), (0,0,0,0,0,-1)) det(J):=((xi(4,3)*xi(3,3)**3 + 2*xi(4,3)*xi(3,3)**2*xi(3,1)*xi(2,4) + xi(4,3)*xi (3,3)*xi(3,1)**2*xi(2,4)**2 - 2*xi(3,3)**3*xi(3,1)*xi(2,4) - xi(3,3)**3*xi(3,1)* xi(2,3) - 2*xi(3,3)**2*xi(3,2)*xi(2,4) - xi(3,3)**2*xi(3,2)*xi(2,3) - 2*xi(3,3) **2*xi(3,1)*xi(1,4) + xi(3,3)*xi(3,2)*xi(3,1)*xi(2,4)**2 - xi(3,3)*xi(3,2)*xi(1, 4) + xi(3,3)*xi(3,1)**2*xi(2,3)*xi(1,4) + xi(3,2)**2*xi(2,4)**2)*xi(4,3) + (xi(3 ,3)**2 + xi(3,2)*xi(2,3))*(xi(3,3)*xi(3,1) + xi(3,2))*xi(1,4))/xi(3,3) matJ_1:= [0 - xi(4,3)*xi(3,3)] [ ] [1 - xi(3,3) ] matJ_3:= [ xi(3,1) xi(3,2) ] [ ] [ xi(3,3)*xi(3,1) + xi(3,2) ] [--------------------------- - xi(4,3)*xi(3,1)] [ xi(3,3) ] matU:= bI+ J_1 matU:= [b - xi(4,3)*xi(3,3)] [ ] [1 - xi(3,3) + b ] matJ_3*matU:= mat((xi(3,2) + xi(3,1)*b, - xi(4,3)*xi(3,3)*xi(3,1) - xi(3,3)*xi(3,2) + xi(3,2)*b), - xi(4,3)*xi(3,3)*xi(3,1) + xi(3,3)*xi(3,1)*b + xi(3,2)*b (------------------------------------------------------------, xi(3,3) - xi(4,3)*(xi(3,2) + xi(3,1)*b))) matU:= [ - xi(3,2) ] [------------ - xi(4,3)*xi(3,3) ] [ xi(3,1) ] [ ] [ xi(3,3)*xi(3,1) + xi(3,2) ] [ 1 - ---------------------------] [ xi(3,1) ] matJ_3:= [ xi(3,1) xi(3,2) ] [ ] [ xi(3,3)*xi(3,1) + xi(3,2) ] [--------------------------- - xi(4,3)*xi(3,1)] [ xi(3,3) ] matJ_3*matU:= 2 2 xi(4,3)*xi(3,3)*xi(3,1) + xi(3,3)*xi(3,2)*xi(3,1) + xi(3,2) mat((0, - ---------------------------------------------------------------), xi(3,1) 2 2 xi(4,3)*xi(3,3)*xi(3,1) + xi(3,3)*xi(3,2)*xi(3,1) + xi(3,2) ( - ---------------------------------------------------------------,0)) xi(3,3)*xi(3,1) 2 2 xi(4,3)*xi(3,3)*xi(3,1) + xi(3,3)*xi(3,2)*xi(3,1) + xi(3,2) det(matU):=--------------------------------------------------------------- 2 xi(3,1) localrecap USD \par Now the nonzero torsion equations left are : {{{{1,2},1},0}, {{{1,2},2},0}, {{{1,2},3},0}, {{{1,2},4},0}, {{{1,2},5},0}, {{{1,2},6},( - xi(6,5)*xi(3,3)**2 + xi(3,2)**2)/xi(3,3)}, {{{1,3},1},0}, {{{1,3},2},0}, {{{1,3},3},0}, {{{1,3},4},0}, {{{1,3},5}, ( - xi(6,5)*xi(4,3)*xi(3,3)**3*xi(2,4) - xi(6,5)*xi(4,3)*xi(3,3)**3*xi(2,3) + xi (6,5)*xi(3,3)**2*xi(2,3) + xi(4,3)**2*xi(3,3)**2*xi(3,2) - 2*xi(4,3)*xi(3,3)*xi( 3,2) + xi(3,3)**2*xi(3,2) + xi(3,2))/(xi(6,5)*xi(3,3)**3)}, {{{1,3},6}, (xi(6,5)*xi(3,3)**2*xi(2,3) - xi(4,3)*xi(3,3)*xi(3,2) + xi(3,3)**2*xi(3,2) + xi( 3,2))/xi(3,3)**2}, {{{1,4},1},0}, {{{1,4},2},0}, {{{1,4},3},0}, {{{1,4},4},0}, {{{1,4},5}, ( - xi(4,3)*xi(3,3)*xi(2,4) + xi(3,3)*xi(1,4) + xi(2,4))/xi(3,3)}, {{{1,4},6},xi(6,5)*xi(2,4) - xi(3,2)}, {{{1,5},1},0}, {{{1,5},2},0}, {{{1,5},3},0}, {{{1,5},4},0}, {{{1,5},5},0}, {{{1,5},6},0}, {{{1,6},1},0}, {{{1,6},2},0}, {{{1,6},3},0}, {{{1,6},4},0}, {{{1,6},5},0}, {{{1,6},6},0}, {{{2,3},1},0}, {{{2,3},2},0}, {{{2,3},3},0}, {{{2,3},4},0}, {{{2,3},5}, (xi(4,3)*( - xi(4,3)*xi(3,3)*xi(2,4) + xi(3,3)**2*xi(2,4) + xi(3,3)**2*xi(2,3) + xi(2,4)))/xi(3,3)}, {{{2,3},6}, xi(4,3)*(xi(6,5)*xi(2,4) - xi(3,2))}, {{{2,4},1},0}, {{{2,4},2},0}, {{{2,4},3},0}, {{{2,4},4},0}, {{{2,4},5}, (xi(6,5)*xi(4,3)*xi(3,3)**3*xi(2,4) + xi(6,5)*xi(4,3)*xi(3,3)**2*xi(1,4) - xi(6, 5)*xi(3,3)**3*xi(1,4) - xi(6,5)*xi(3,3)*xi(1,4) - xi(4,3)**2*xi(3,3)**2*xi(3,2) + 2*xi(4,3)*xi(3,3)*xi(3,2) - xi(3,3)**2*xi(3,2) - xi(3,2))/(xi(6,5)*xi(3,3)**2) }, {{{2,4},6}, ( - xi(6,5)*xi(3,3)*xi(1,4) + xi(4,3)*xi(3,3)*xi(3,2) - xi(3,2))/xi(3,3)}, {{{2,5},1},0}, {{{2,5},2},0}, {{{2,5},3},0}, {{{2,5},4},0}, {{{2,5},5},0}, {{{2,5},6},0}, {{{2,6},1},0}, {{{2,6},2},0}, {{{2,6},3},0}, {{{2,6},4},0}, {{{2,6},5},0}, {{{2,6},6},0}, {{{3,4},1},0}, {{{3,4},2},0}, {{{3,4},3},0}, {{{3,4},4},0}, {{{3,4},5}, (xi(6,5)*xi(4,3)*xi(3,3)*xi(2,4)**2 + xi(6,5)*xi(3,3)*xi(2,3)*xi(1,4) - xi(4,3) **2*xi(3,3)**2 + 2*xi(4,3)*xi(3,3) - xi(3,3)**2 - 1)/(xi(6,5)*xi(3,3))}, {{{3,4},6},0}, {{{3,5},1},0}, {{{3,5},2},0}, {{{3,5},3},0}, {{{3,5},4},0}, {{{3,5},5},0}, {{{3,5},6},0}, {{{3,6},1},0}, {{{3,6},2},0}, {{{3,6},3},0}, {{{3,6},4},0}, {{{3,6},5},0}, {{{3,6},6},0}, {{{4,5},1},0}, {{{4,5},2},0}, {{{4,5},3},0}, {{{4,5},4},0}, {{{4,5},5},0}, {{{4,5},6},0}, {{{4,6},1},0}, {{{4,6},2},0}, {{{4,6},3},0}, {{{4,6},4},0}, {{{4,6},5},0}, {{{4,6},6},0}, {{{5,6},5},0}, {{{5,6},6},0}} \par The matrix USD J USD is :\\ USD J^1_1=0;USD\\ USD J^1_2= - xi(4,3)*xi(3,3);USD\\ USD J^1_3= - xi(4,3)*xi(2,4);USD\\ USD J^1_4=xi(1,4);USD\\ USD J^1_5=0;USD\\ USD J^1_6=0;USD\\ USD J^2_1=1;USD\\ USD J^2_2= - xi(3,3);USD\\ USD J^2_3=xi(2,3);USD\\ USD J^2_4=xi(2,4);USD\\ USD J^2_5=0;USD\\ USD J^2_6=0;USD\\ USD J^3_1=0;USD\\ USD J^3_2=xi(3,2);USD\\ USD J^3_3=xi(3,3);USD\\ USD J^3_4= - xi(3,3);USD\\ USD J^3_5=0;USD\\ USD J^3_6=0;USD\\ USD J^4_1=xi(3,2)/xi(3,3);USD\\ USD J^4_2=0;USD\\ USD J^4_3=xi(4,3);USD\\ USD J^4_4=0;USD\\ USD J^4_5=0;USD\\ USD J^4_6=0;USD\\ USD J^5_1=0;USD\\ USD J^5_2=0;USD\\ USD J^5_3=0;USD\\ USD J^5_4=0;USD\\ USD J^5_5=( - xi(4,3)*xi(3,3) + 1)/xi(3,3);USD\\ USD J^5_6=( - xi(4,3)**2*xi(3,3)**2 + 2*xi(4,3)*xi(3,3) - xi(3,3)**2 - 1)/(xi(6 ,5)*xi(3,3)**2);USD\\ USD J^6_1=0;USD\\ USD J^6_2=0;USD\\ USD J^6_3=0;USD\\ USD J^6_4=0;USD\\ USD J^6_5=xi(6,5);USD\\ USD J^6_6=(xi(4,3)*xi(3,3) - 1)/xi(3,3);USD\\ \\USD J^2(1,1):=( - xi(4,3)*xi(3,3)**2 + xi(3,2)*xi(1,4))/xi(3,3)USD\\ \\USD J^2(1,2):=xi(4,3)*(xi(3,3)**2 - xi(3,2)*xi(2,4))USD\\ \\USD J^2(1,3):=xi(4,3)*( - xi(3,3)*xi(2,4) - xi(3,3)*xi(2,3) + xi(1,4))USD\\ \\USD J^2(1,4):=0USD\\ \\USD J^2(1,5):=0USD\\ \\USD J^2(1,6):=0USD\\ \\USD J^2(2,1):=( - xi(3,3)**2 + xi(3,2)*xi(2,4))/xi(3,3)USD\\ \\USD J^2(2,2):= - xi(4,3)*xi(3,3) + xi(3,3)**2 + xi(3,2)*xi(2,3)USD\\ \\USD J^2(2,3):=0USD\\ \\USD J^2(2,4):= - xi(3,3)*xi(2,4) - xi(3,3)*xi(2,3) + xi(1,4)USD\\ \\USD J^2(2,5):=0USD\\ \\USD J^2(2,6):=0USD\\ \\USD J^2(3,1):=0USD\\ \\USD J^2(3,2):=0USD\\ \\USD J^2(3,3):= - xi(4,3)*xi(3,3) + xi(3,3)**2 + xi(3,2)*xi(2,3)USD\\ \\USD J^2(3,4):= - xi(3,3)**2 + xi(3,2)*xi(2,4)USD\\ \\USD J^2(3,5):=0USD\\ \\USD J^2(3,6):=0USD\\ \\USD J^2(4,1):=0USD\\ \\USD J^2(4,2):=0USD\\ \\USD J^2(4,3):=(xi(4,3)*(xi(3,3)**2 - xi(3,2)*xi(2,4)))/xi(3,3)USD\\ \\USD J^2(4,4):=( - xi(4,3)*xi(3,3)**2 + xi(3,2)*xi(1,4))/xi(3,3)USD\\ \\USD J^2(4,5):=0USD\\ \\USD J^2(4,6):=0USD\\ \\USD J^2(5,1):=0USD\\ \\USD J^2(5,2):=0USD\\ \\USD J^2(5,3):=0USD\\ \\USD J^2(5,4):=0USD\\ \\USD J^2(5,5):=-1USD\\ \\USD J^2(5,6):=0USD\\ \\USD J^2(6,1):=0USD\\ \\USD J^2(6,2):=0USD\\ \\USD J^2(6,3):=0USD\\ \\USD J^2(6,4):=0USD\\ \\USD J^2(6,5):=0USD\\ \\USD J^2(6,6):=-1USD\\ Trace(J):=0 J:= mat((0, - xi(4,3)*xi(3,3), - xi(4,3)*xi(2,4),xi(1,4),0,0), (1, - xi(3,3),xi(2,3),xi(2,4),0,0), (0,xi(3,2),xi(3,3), - xi(3,3),0,0), xi(3,2) (---------,0,xi(4,3),0,0,0), xi(3,3) - xi(4,3)*xi(3,3) + 1 (0,0,0,0,------------------------, xi(3,3) 2 2 2 - xi(4,3) *xi(3,3) + 2*xi(4,3)*xi(3,3) - xi(3,3) - 1 ---------------------------------------------------------), 2 xi(6,5)*xi(3,3) xi(4,3)*xi(3,3) - 1 (0,0,0,0,xi(6,5),---------------------)) xi(3,3) J**2:= 2 - xi(4,3)*xi(3,3) + xi(3,2)*xi(1,4) mat((---------------------------------------, xi(3,3) 2 xi(4,3)*(xi(3,3) - xi(3,2)*xi(2,4)), xi(4,3)*( - xi(3,3)*xi(2,4) - xi(3,3)*xi(2,3) + xi(1,4)),0,0,0), 2 - xi(3,3) + xi(3,2)*xi(2,4) (-------------------------------, xi(3,3) 2 - xi(4,3)*xi(3,3) + xi(3,3) + xi(3,2)*xi(2,3),0, - xi(3,3)*xi(2,4) - xi(3,3)*xi(2,3) + xi(1,4),0,0), 2 (0,0, - xi(4,3)*xi(3,3) + xi(3,3) + xi(3,2)*xi(2,3), 2 - xi(3,3) + xi(3,2)*xi(2,4),0,0), 2 xi(4,3)*(xi(3,3) - xi(3,2)*xi(2,4)) (0,0,--------------------------------------, xi(3,3) 2 - xi(4,3)*xi(3,3) + xi(3,2)*xi(1,4) ---------------------------------------,0,0), xi(3,3) (0,0,0,0,-1,0), (0,0,0,0,0,-1)) det(J):=((xi(4,3)*xi(3,3)**3 - 2*xi(3,3)**2*xi(3,2)*xi(2,4) - xi(3,3)**2*xi(3,2) *xi(2,3) - xi(3,3)*xi(3,2)*xi(1,4) + xi(3,2)**2*xi(2,4)**2)*xi(4,3) + (xi(3,3)** 2 + xi(3,2)*xi(2,3))*xi(3,2)*xi(1,4))/xi(3,3) Then from the equations %{{{1,4},6},xi(6,5)*xi(2,4) - xi(3,2)}, \\ USD xi(6,5):=xi(3,2)/xi(2,4)USD %\\USD J^2(2,1):=( - xi(3,3)**2 + xi(3,2)*xi(2,4))/xi(3,3)USD\\ \\ USD xi(2,4):=xi(3,3)**2/xi(3,2)USD %{{{1,4},5}, ( - xi(4,3)*xi(3,3)**2 + xi(3,3) + xi(3,2)*xi(1,4))/xi(3,2)}, \\ USD xi(1,4):=(xi(3,3)*(xi(4,3)*xi(3,3) - 1))/xi(3,2)USD !%{{{1,3},6},! ! (xi(3,2)*(! -! xi(4,3)*xi(3,3)! +! xi(3,3)**2! +! xi(3,2)*xi(2, 3)! +! 1)\ )/xi(3,3)**2},! ! \\ USD xi(2,3):=(xi(4,3)*xi(3,3) - xi(3,3)**2 - 1)/xi(3,2)USD localrecap USD \par Now the nonzero torsion equations left are : {{{{1,2},1},0}, {{{1,2},2},0}, {{{1,2},3},0}, {{{1,2},4},0}, {{{1,2},5},0}, {{{1,2},6},0}, {{{1,3},1},0}, {{{1,3},2},0}, {{{1,3},3},0}, {{{1,3},4},0}, {{{1,3},5},0}, {{{1,3},6},0}, {{{1,4},1},0}, {{{1,4},2},0}, {{{1,4},3},0}, {{{1,4},4},0}, {{{1,4},5},0}, {{{1,4},6},0}, {{{1,5},1},0}, {{{1,5},2},0}, {{{1,5},3},0}, {{{1,5},4},0}, {{{1,5},5},0}, {{{1,5},6},0}, {{{1,6},1},0}, {{{1,6},2},0}, {{{1,6},3},0}, {{{1,6},4},0}, {{{1,6},5},0}, {{{1,6},6},0}, {{{2,3},1},0}, {{{2,3},2},0}, {{{2,3},3},0}, {{{2,3},4},0}, {{{2,3},5},0}, {{{2,3},6},0}, {{{2,4},1},0}, {{{2,4},2},0}, {{{2,4},3},0}, {{{2,4},4},0}, {{{2,4},5},0}, {{{2,4},6},0}, {{{2,5},1},0}, {{{2,5},2},0}, {{{2,5},3},0}, {{{2,5},4},0}, {{{2,5},5},0}, {{{2,5},6},0}, {{{2,6},1},0}, {{{2,6},2},0}, {{{2,6},3},0}, {{{2,6},4},0}, {{{2,6},5},0}, {{{2,6},6},0}, {{{3,4},1},0}, {{{3,4},2},0}, {{{3,4},3},0}, {{{3,4},4},0}, {{{3,4},5},0}, {{{3,4},6},0}, {{{3,5},1},0}, {{{3,5},2},0}, {{{3,5},3},0}, {{{3,5},4},0}, {{{3,5},5},0}, {{{3,5},6},0}, {{{3,6},1},0}, {{{3,6},2},0}, {{{3,6},3},0}, {{{3,6},4},0}, {{{3,6},5},0}, {{{3,6},6},0}, {{{4,5},1},0}, {{{4,5},2},0}, {{{4,5},3},0}, {{{4,5},4},0}, {{{4,5},5},0}, {{{4,5},6},0}, {{{4,6},1},0}, {{{4,6},2},0}, {{{4,6},3},0}, {{{4,6},4},0}, {{{4,6},5},0}, {{{4,6},6},0}, {{{5,6},5},0}, {{{5,6},6},0}} \par The matrix USD J USD is :\\ USD J^1_1=0;USD\\ USD J^1_2= - xi(4,3)*xi(3,3);USD\\ USD J^1_3=( - xi(4,3)*xi(3,3)**2)/xi(3,2);USD\\ USD J^1_4=(xi(3,3)*(xi(4,3)*xi(3,3) - 1))/xi(3,2);USD\\ USD J^1_5=0;USD\\ USD J^1_6=0;USD\\ USD J^2_1=1;USD\\ USD J^2_2= - xi(3,3);USD\\ USD J^2_3=(xi(4,3)*xi(3,3) - xi(3,3)**2 - 1)/xi(3,2);USD\\ USD J^2_4=xi(3,3)**2/xi(3,2);USD\\ USD J^2_5=0;USD\\ USD J^2_6=0;USD\\ USD J^3_1=0;USD\\ USD J^3_2=xi(3,2);USD\\ USD J^3_3=xi(3,3);USD\\ USD J^3_4= - xi(3,3);USD\\ USD J^3_5=0;USD\\ USD J^3_6=0;USD\\ USD J^4_1=xi(3,2)/xi(3,3);USD\\ USD J^4_2=0;USD\\ USD J^4_3=xi(4,3);USD\\ USD J^4_4=0;USD\\ USD J^4_5=0;USD\\ USD J^4_6=0;USD\\ USD J^5_1=0;USD\\ USD J^5_2=0;USD\\ USD J^5_3=0;USD\\ USD J^5_4=0;USD\\ USD J^5_5=( - xi(4,3)*xi(3,3) + 1)/xi(3,3);USD\\ USD J^5_6=( - xi(4,3)**2*xi(3,3)**2 + 2*xi(4,3)*xi(3,3) - xi(3,3)**2 - 1)/xi(3, 2)**2;USD\\ USD J^6_1=0;USD\\ USD J^6_2=0;USD\\ USD J^6_3=0;USD\\ USD J^6_4=0;USD\\ USD J^6_5=xi(3,2)**2/xi(3,3)**2;USD\\ USD J^6_6=(xi(4,3)*xi(3,3) - 1)/xi(3,3);USD\\ \\USD J^2(1,1):=-1USD\\ \\USD J^2(1,2):=0USD\\ \\USD J^2(1,3):=0USD\\ \\USD J^2(1,4):=0USD\\ \\USD J^2(1,5):=0USD\\ \\USD J^2(1,6):=0USD\\ \\USD J^2(2,1):=0USD\\ \\USD J^2(2,2):=-1USD\\ \\USD J^2(2,3):=0USD\\ \\USD J^2(2,4):=0USD\\ \\USD J^2(2,5):=0USD\\ \\USD J^2(2,6):=0USD\\ \\USD J^2(3,1):=0USD\\ \\USD J^2(3,2):=0USD\\ \\USD J^2(3,3):=-1USD\\ \\USD J^2(3,4):=0USD\\ \\USD J^2(3,5):=0USD\\ \\USD J^2(3,6):=0USD\\ \\USD J^2(4,1):=0USD\\ \\USD J^2(4,2):=0USD\\ \\USD J^2(4,3):=0USD\\ \\USD J^2(4,4):=-1USD\\ \\USD J^2(4,5):=0USD\\ \\USD J^2(4,6):=0USD\\ \\USD J^2(5,1):=0USD\\ \\USD J^2(5,2):=0USD\\ \\USD J^2(5,3):=0USD\\ \\USD J^2(5,4):=0USD\\ \\USD J^2(5,5):=-1USD\\ \\USD J^2(5,6):=0USD\\ \\USD J^2(6,1):=0USD\\ \\USD J^2(6,2):=0USD\\ \\USD J^2(6,3):=0USD\\ \\USD J^2(6,4):=0USD\\ \\USD J^2(6,5):=0USD\\ \\USD J^2(6,6):=-1USD\\ Trace(J):=0 J:= 2 - xi(4,3)*xi(3,3) xi(3,3)*(xi(4,3)*xi(3,3) - 1) mat((0, - xi(4,3)*xi(3,3),---------------------,-------------------------------, xi(3,2) xi(3,2) 0,0), 2 2 xi(4,3)*xi(3,3) - xi(3,3) - 1 xi(3,3) (1, - xi(3,3),--------------------------------,----------,0,0), xi(3,2) xi(3,2) (0,xi(3,2),xi(3,3), - xi(3,3),0,0), xi(3,2) (---------,0,xi(4,3),0,0,0), xi(3,3) - xi(4,3)*xi(3,3) + 1 (0,0,0,0,------------------------, xi(3,3) 2 2 2 - xi(4,3) *xi(3,3) + 2*xi(4,3)*xi(3,3) - xi(3,3) - 1 ---------------------------------------------------------), 2 xi(3,2) 2 xi(3,2) xi(4,3)*xi(3,3) - 1 (0,0,0,0,----------,---------------------)) 2 xi(3,3) xi(3,3) J**2:= [-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] det(J):=1 Finally, with a Phi having diag(bI,I) with suitable b=xi(3,3)/xi(3,2), one may suppose xi(3,2)=xi(3,3) \\ USD xi(3,2):=xi(3,3)USD localrecap USD \par Now the nonzero torsion equations left are : {{{{1,2},1},0}, {{{1,2},2},0}, {{{1,2},3},0}, {{{1,2},4},0}, {{{1,2},5},0}, {{{1,2},6},0}, {{{1,3},1},0}, {{{1,3},2},0}, {{{1,3},3},0}, {{{1,3},4},0}, {{{1,3},5},0}, {{{1,3},6},0}, {{{1,4},1},0}, {{{1,4},2},0}, {{{1,4},3},0}, {{{1,4},4},0}, {{{1,4},5},0}, {{{1,4},6},0}, {{{1,5},1},0}, {{{1,5},2},0}, {{{1,5},3},0}, {{{1,5},4},0}, {{{1,5},5},0}, {{{1,5},6},0}, {{{1,6},1},0}, {{{1,6},2},0}, {{{1,6},3},0}, {{{1,6},4},0}, {{{1,6},5},0}, {{{1,6},6},0}, {{{2,3},1},0}, {{{2,3},2},0}, {{{2,3},3},0}, {{{2,3},4},0}, {{{2,3},5},0}, {{{2,3},6},0}, {{{2,4},1},0}, {{{2,4},2},0}, {{{2,4},3},0}, {{{2,4},4},0}, {{{2,4},5},0}, {{{2,4},6},0}, {{{2,5},1},0}, {{{2,5},2},0}, {{{2,5},3},0}, {{{2,5},4},0}, {{{2,5},5},0}, {{{2,5},6},0}, {{{2,6},1},0}, {{{2,6},2},0}, {{{2,6},3},0}, {{{2,6},4},0}, {{{2,6},5},0}, {{{2,6},6},0}, {{{3,4},1},0}, {{{3,4},2},0}, {{{3,4},3},0}, {{{3,4},4},0}, {{{3,4},5},0}, {{{3,4},6},0}, {{{3,5},1},0}, {{{3,5},2},0}, {{{3,5},3},0}, {{{3,5},4},0}, {{{3,5},5},0}, {{{3,5},6},0}, {{{3,6},1},0}, {{{3,6},2},0}, {{{3,6},3},0}, {{{3,6},4},0}, {{{3,6},5},0}, {{{3,6},6},0}, {{{4,5},1},0}, {{{4,5},2},0}, {{{4,5},3},0}, {{{4,5},4},0}, {{{4,5},5},0}, {{{4,5},6},0}, {{{4,6},1},0}, {{{4,6},2},0}, {{{4,6},3},0}, {{{4,6},4},0}, {{{4,6},5},0}, {{{4,6},6},0}, {{{5,6},5},0}, {{{5,6},6},0}} \par The matrix USD J USD is :\\ USD J^1_1=0;USD\\ USD J^1_2= - xi(4,3)*xi(3,3);USD\\ USD J^1_3= - xi(4,3)*xi(3,3);USD\\ USD J^1_4=xi(4,3)*xi(3,3) - 1;USD\\ USD J^1_5=0;USD\\ USD J^1_6=0;USD\\ USD J^2_1=1;USD\\ USD J^2_2= - xi(3,3);USD\\ USD J^2_3=(xi(4,3)*xi(3,3) - xi(3,3)**2 - 1)/xi(3,3);USD\\ USD J^2_4=xi(3,3);USD\\ USD J^2_5=0;USD\\ USD J^2_6=0;USD\\ USD J^3_1=0;USD\\ USD J^3_2=xi(3,3);USD\\ USD J^3_3=xi(3,3);USD\\ USD J^3_4= - xi(3,3);USD\\ USD J^3_5=0;USD\\ USD J^3_6=0;USD\\ USD J^4_1=1;USD\\ USD J^4_2=0;USD\\ USD J^4_3=xi(4,3);USD\\ USD J^4_4=0;USD\\ USD J^4_5=0;USD\\ USD J^4_6=0;USD\\ USD J^5_1=0;USD\\ USD J^5_2=0;USD\\ USD J^5_3=0;USD\\ USD J^5_4=0;USD\\ USD J^5_5=( - xi(4,3)*xi(3,3) + 1)/xi(3,3);USD\\ USD J^5_6=( - xi(4,3)**2*xi(3,3)**2 + 2*xi(4,3)*xi(3,3) - xi(3,3)**2 - 1)/xi(3, 3)**2;USD\\ USD J^6_1=0;USD\\ USD J^6_2=0;USD\\ USD J^6_3=0;USD\\ USD J^6_4=0;USD\\ USD J^6_5=1;USD\\ USD J^6_6=(xi(4,3)*xi(3,3) - 1)/xi(3,3);USD\\ \\USD J^2(1,1):=-1USD\\ \\USD J^2(1,2):=0USD\\ \\USD J^2(1,3):=0USD\\ \\USD J^2(1,4):=0USD\\ \\USD J^2(1,5):=0USD\\ \\USD J^2(1,6):=0USD\\ \\USD J^2(2,1):=0USD\\ \\USD J^2(2,2):=-1USD\\ \\USD J^2(2,3):=0USD\\ \\USD J^2(2,4):=0USD\\ \\USD J^2(2,5):=0USD\\ \\USD J^2(2,6):=0USD\\ \\USD J^2(3,1):=0USD\\ \\USD J^2(3,2):=0USD\\ \\USD J^2(3,3):=-1USD\\ \\USD J^2(3,4):=0USD\\ \\USD J^2(3,5):=0USD\\ \\USD J^2(3,6):=0USD\\ \\USD J^2(4,1):=0USD\\ \\USD J^2(4,2):=0USD\\ \\USD J^2(4,3):=0USD\\ \\USD J^2(4,4):=-1USD\\ \\USD J^2(4,5):=0USD\\ \\USD J^2(4,6):=0USD\\ \\USD J^2(5,1):=0USD\\ \\USD J^2(5,2):=0USD\\ \\USD J^2(5,3):=0USD\\ \\USD J^2(5,4):=0USD\\ \\USD J^2(5,5):=-1USD\\ \\USD J^2(5,6):=0USD\\ \\USD J^2(6,1):=0USD\\ \\USD J^2(6,2):=0USD\\ \\USD J^2(6,3):=0USD\\ \\USD J^2(6,4):=0USD\\ \\USD J^2(6,5):=0USD\\ \\USD J^2(6,6):=-1USD\\ Trace(J):=0 J:= mat((0, - xi(4,3)*xi(3,3), - xi(4,3)*xi(3,3),xi(4,3)*xi(3,3) - 1,0,0), 2 xi(4,3)*xi(3,3) - xi(3,3) - 1 (1, - xi(3,3),--------------------------------,xi(3,3),0,0), xi(3,3) (0,xi(3,3),xi(3,3), - xi(3,3),0,0), (1,0,xi(4,3),0,0,0), - xi(4,3)*xi(3,3) + 1 (0,0,0,0,------------------------, xi(3,3) 2 2 2 - xi(4,3) *xi(3,3) + 2*xi(4,3)*xi(3,3) - xi(3,3) - 1 ---------------------------------------------------------), 2 xi(3,3) xi(4,3)*xi(3,3) - 1 (0,0,0,0,1,---------------------)) xi(3,3) J**2:= [-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] det(J):=1 \\ Then USD J USD has entries : USD J^1_1=0;USD\\ USD J^1_2= - xi(4,3)*xi(3,3);USD\\ USD J^1_3= - xi(4,3)*xi(3,3);USD\\ USD J^1_4=xi(4,3)*xi(3,3) - 1;USD\\ USD J^1_5=0;USD\\ USD J^1_6=0;USD\\ USD J^2_1=1;USD\\ USD J^2_2= - xi(3,3);USD\\ USD J^2_3=( - (xi(3,3)**2 + 1 - xi(4,3)*xi(3,3)))/xi(3,3);USD\\ USD J^2_4=xi(3,3);USD\\ USD J^2_5=0;USD\\ USD J^2_6=0;USD\\ USD J^3_1=0;USD\\ USD J^3_2=xi(3,3);USD\\ USD J^3_3=xi(3,3);USD\\ USD J^3_4= - xi(3,3);USD\\ USD J^3_5=0;USD\\ USD J^3_6=0;USD\\ USD J^4_1=1;USD\\ USD J^4_2=0;USD\\ USD J^4_3=xi(4,3);USD\\ USD J^4_4=0;USD\\ USD J^4_5=0;USD\\ USD J^4_6=0;USD\\ USD J^5_1=0;USD\\ USD J^5_2=0;USD\\ USD J^5_3=0;USD\\ USD J^5_4=0;USD\\ USD J^5_5=( - (xi(4,3)*xi(3,3) - 1))/xi(3,3);USD\\ USD J^5_6=( - ((xi(4,3)*xi(3,3) - 2)*xi(4,3)*xi(3,3) + xi(3,3)**2 + 1))/xi(3,3) **2;USD\\ USD J^6_1=0;USD\\ USD J^6_2=0;USD\\ USD J^6_3=0;USD\\ USD J^6_4=0;USD\\ USD J^6_5=1;USD\\ USD J^6_6=(xi(4,3)*xi(3,3) - 1)/xi(3,3);USD\\ Hence we are finally reduced to \\ {\fontsize{8}{10} \selectfont USDUSD J(\xi^3_3,\xi^4_3) := \begin{pmatrix} 0& - xi(4,3)*xi(3,3)& - xi(4,3)*xi(3,3)& xi(4,3)*xi(3,3) - 1& 0& 0\\ 1& - xi(3,3)& ( - (xi(3,3)**2 + 1 - xi(4,3)*xi(3,3)))/xi(3,3)& xi(3,3)& 0& 0\\ 0& xi(3,3)& xi(3,3)& - xi(3,3)& 0& 0\\ 1& 0& xi(4,3)& 0& 0& 0\\ 0& 0& 0& 0& ( - (xi(4,3)*xi(3,3) - 1))/xi(3,3)& ( - ((xi(4,3)*xi(3,3) - 2)*xi(4,3)*xi(3,3) + xi(3,3)**2 + 1))/xi(3,3)**2\\ 0& 0& 0& 0& 1& (xi(4,3)*xi(3,3) - 1)/xi(3,3)\end{pmatrix}USDUSD } with condition USD xi(3,3) \neq 0 USD. \par Commutation relations of USD \mathfrak{m} : USD USD[\tilde{x}_1,\tilde{x}_2]=xi(3,3)*tildex_6 + tildex_5 - xi(4,3)*xi(3,3)* tildex_5USD; USD[\tilde{x}_1,\tilde{x}_3]= - (xi(4,3)*tildex_5 - tildex_6)*xi(3,3)USD; USD[\tilde{x}_1,\tilde{x}_4]= - (xi(3,3)*tildex_6 + tildex_5 - xi(4,3)*xi(3,3)* tildex_5)USD; USD[\tilde{x}_2,\tilde{x}_3]= - (xi(3,3)*tildex_6 + tildex_5 - xi(4,3)*xi(3,3)* tildex_5)*xi(4,3)USD; USD[\tilde{x}_2,\tilde{x}_4]=xi(3,3)*tildex_5USD; USD[\tilde{x}_3,\tilde{x}_4]=( - ((xi(4,3)*xi(3,3) - 1)*xi(3,3)*tildex_6 - (xi(3 ,3)**2 + 1)*tildex_5 - (xi(4,3)*xi(3,3) - 2)*xi(4,3)*xi(3,3)*tildex_5))/xi(3,3) USD; \P \par Now we check if the condition USD [Jx,y]= J[x,y] USD is satisfied USD\forall x,y \in {\mathcal{G}}_{6,n2},USD \textit{i.e.} if USD{\mathcal{G}}_{6,n2}USD is a \textit{complex} algebra. \\USD J[x_j,x_k] \neq [Jx_j,x_k] USD in the following cases{{{1,1}, - x(5)}, {{1,2}, - (x(6)*xi(3,3) - x(5)*(xi(4,3)*xi(3,3) - 1))/xi(3,3)}, {{1,3}, - x(6)}, {{2,1}, x(6) - ((xi(4,3)*xi(3,3) - xi(3,3)**2 - 1)*x(5))/xi(3,3)}, {{2,2}, - xi(4,3)*xi(3,3)*x(5)}, {{2,4},xi(3,3)*x(6)}, {{3,1}, - ((xi(4,3)*xi(3,3) - xi(3,3)**2 - 1)*x(5))/xi(3,3)}, {{3,2}, - xi(4,3)*xi(3,3)*x(5)}, {{3,3}, - xi(4,3)*x(6)}, {{3,4}, (x(6)*(xi(3,3)**2 + 1 - xi(4,3)*xi(3,3)))/xi(3,3) + ((xi(4,3)**2*xi(3,3)**2 - 2* xi(4,3)*xi(3,3) + xi(3,3)**2 + 1)*x(5))/xi(3,3)**2}, {{4,1}, - xi(3,3)*x(5)}, {{4,2},(xi(4,3)*xi(3,3) - 1)*x(5)}, {{4,3}, (x(6)*(xi(4,3)*xi(3,3) - 1))/xi(3,3) - ((xi(4,3)**2*xi(3,3)**2 - 2*xi(4,3)*xi(3, 3) + xi(3,3)**2 + 1)*x(5))/xi(3,3)**2}, {{4,4}, - xi(3,3)*x(6)}} \end{document}