%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%This is program6_6.red  formerly structcompl6_1tex13.tex
%computes all complex structures on the Lie algebra g_{6,1} in the case
% xi(2,3) neq 0.
%%%%%%%%%%%%%%  output   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   off echo$
    off nat$
   OUT  "rprogram6_6.tex"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%  loading the commutation relations file%%%%%%%%%%%%%%%%%%
DIM:= 6$
  in "6nilp/6nilp.1"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%writing the commutation relations
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
WRITE "\documentclass{article}"$
WRITE "\usepackage{amsmath,amssymb}"$
WRITE "\sloppy"$
WRITE "\begin{document}"$
%WRITE "This output from the file \texttt{structcompl6\_1tex13.tex}.\\"$
WRITE "This output from the file \texttt{rprogram6_6.tex}.\\"$
WRITE "Computation of all complex    structures on the real Lie Algebra"$
write "USD {\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}.USD"$
write "Case USD xi(2,3) \neq 0 USD."$
WRITE "\smallskip  \par "$
WRITE "Commutation relations for"$
write "USD {\mathcal{G}}_{", dim,",", PART(REFALGTEX,1), "}:USD\\"$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO X_(i,j):=X(i)*x(j)$
%FOR j:=1:DIM DO X(j):=MKID(x_,j)$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO
    IF X_(i,j) NEQ 0 THEN WRITE
 %       "USD[x_",i,",x_",j,"]=", X_(i,j),"USD;"$
        "USD[x(",i,"),x(",j,")]=", X_(i,j),"USD;"$
WRITE "\P"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 operator x,f,A$
 noncom x,f$
 for all j let
  F(X(j))=FOR j1:=1:DIM SUM xi(j1,j)*X(j1)$
  let f(0)=0$
 for all j let f(-x(j))=-f(x(j))$
 FOR ALL J,U SUCH THAT NUMBERP(U) LET f(U*X(j))=U*f(X(j))$
 FOR ALL J,U,V SUCH THAT NUMBERP(U) AND NUMBERP(V) LET f(U*V*X(j))=U*V*f(X(j))$
 FOR ALL J,U,V SUCH THAT NUMBERP(U) AND NUMBERP(V) LET f(-U*V*X(j))=-U*V*f(X(j))$
 FOR ALL s,i,j,k,l
 LET f(X(s) *xi(i,j)*xi(k,l))=xi(i,j)*xi(k,l)*f(X(s))$
 FOR ALL s,i,j,k,l
 LET f(xi(i,j)*xi(k,l)*X(s))=xi(i,j)*xi(k,l)*f(X(s))$
 FOR ALL s,i,j,k,l
 LET f(-X(s) *xi(i,j)*xi(k,l))=-xi(i,j)*xi(k,l)*f(X(s))$
 FOR ALL s,i,j
 LET f(X(s) *xi(i,j))=(xi(i,j))*f(X(s))$
 FOR ALL s,i,j
 LET f(-X(s) *xi(i,j))=-(xi(i,j))*f(X(s))$
 FOR ALL s,i,j
 LET f(-X(s) *xi(i,j)**2)=(-xi(i,j)**2)*f(X(s))$
 FOR ALL s,i,j
 LET f(X(s) *xi(i,j)**2)=(xi(i,j)**2)*f(X(s))$
 FOR ALL s,i,j,k,l
 LET f(-xi(i,j)*xi(k,l)*X(s))=-xi(i,j)*xi(k,l)*f(X(s))$
 FOR ALL J,U SUCH THAT NUMBERP(U) LET f(-U*X(j))=-U*f(X(j))$
 FOR ALL A,B LET F(A+B)=F(A)+F(B)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 for all j,k let
  temp110(j,k) = x(j)*F(x(k))$
 for all j,k let
  temp120(j,k) = F(temp110(j,k))$
 for all j,k let
  temp101(j,k) = F(x(j))*x(k)$
 for all j,k let
  temp102(j,k) = F(temp101(j,k))$
 for all j,k let
  temp2(j,k) =    F(x(j))* F(x(k)) $
 for all j,k let  temp0(j,k) =  x(j)*x(k)$
 for all j,k let
  A(j,k)=  -temp2(j,k) + temp0(j,k) + temp120(j,k) + temp102(j,k);

COLLECT_TORSION:=FOR j1:=1:DIM-1 JOIN
   FOR j2:=j1+1:DIM JOIN
   IF A(j1,j2) NEQ 0 THEN
   {{{j1,j2},A(j1,j2)}}
   ELSE {}$

%IF LENGTH(COLLECT_TORSION) NEQ 0 THEN WRITE "Nonzero torsion
%   in the following cases :",COLLECT_TORSION

IF LENGTH(COLLECT_TORSION) NEQ 0 THEN WRITE "Nonzero torsion"
ELSE WRITE "Zero torsion"$

write "\par"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off exp$
on factor$

%la liste des équations non nulles
%write "list of the nonzero torsion equations"$
COLLECT_EQ:=FOR j1:=1:LENGTH(COLLECT_TORSION) JOIN
   FOR j2:=1:DIM JOIN
   IF V(j2)*PART(PART(COLLECT_TORSION,j1),2) NEQ 0 THEN
   {{{PART(PART(COLLECT_TORSION,j1),1),j2}, V(j2)*PART(PART(COLLECT_TORSION,j1),2)} }
    ELSE {}$

%COMMENT
% WRITE "Torsion equations to cancel (Reduce output) : \\", COLLECT_EQ$

 WRITE "Torsion equations to cancel (Latex output) : \\USD"$
 for each A in COLLECT_EQ do
        if PART(A,2) neq 0 then
        <<write PART(PART(A,1),1),"|", PART(PART(A,1),2),"\\",PART(A,2),"\\">>   $
        %<<write "\\", A,"\\">>   $
 write "USD"$


 off factor$
 on exp$
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par Simultaneous resolution of the nonzero torsion equations and the matrix"$
write "equation USD J^2 = -I . USD"$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ One first gets"$
write "\\ from equation USD36|2USD :"$
xi(2,6):=0$
write "\\ USD xi(2,6):=", xi(2,6),"USD"$
write "\\ and from equation USD46|1USD :"$
xi(1,6):=0$
write "\\ USD xi(1,6):=", xi(1,6),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "then, one gets from  equation USD15|2USD :"$
xi(2,5):=0$
write "\\ USD xi(2,5):=", xi(2,5),"USD"$
write "\\ and from equation USD16|4USD :"$
xi(4,6):=0$
write "\\ USD xi(4,6):=", xi(4,6),"USD"$
write "\\ and from equation USD25|1USD :"$
xi(1,5):=0$
write "\\ USD xi(1,5):=", xi(1,5),"USD"$
write "\\ and from equation USD26|3USD :"$
xi(3,6):=0$
write "\\ USD xi(3,6):=", xi(3,6),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 6x6 entry in  USD J**2 USD  is "$
write "USD  J**2^6_6=xi(6,6)**2 + xi(6,5)*xi(5,6);USD\\"$
write "Hence USD xi(5,6) \neq 0. USD"$
%{{{1,5},5},xi(5,6)*xi(4,5)},
write "then, one gets from  equation USD15|5USD :"$
xi(4,5):=0$
write "\\ USD xi(4,5):=", ws,"USD"$
write "\\ and from equation USD25|5USD :"$
%{{{2,5},5},xi(5,6)*xi(3,5)},
xi(3,5):=0$
write "\\ USD xi(3,5):=", ws,"USD"$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 6x5 entry in  USD J**2 USD  is "$
write "USD  J**2^6_5=xi(6,5)*(xi(6,6) + xi(5,5));USD\\"$
write "Hence as USD xi(5,6) \neq 0, USD"$
write "one gets USD xi(6,6)=-xi(5,5) , USD"$
xi(6,6):=-xi(5,5)$
write "\\ USD xi(6,6):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 6x6 entry in  USD J**2 USD  is "$
write "USD  J**2^6_6=xi(5,5)**2 + xi(6,5)*xi(5,6);USD\\"$
write "Hence USD xi(5,6) =(-1-xi(5,5)**2)/xi(6,5). USD"$
xi(5,6):= (-1-xi(5,5)**2)/xi(6,5)$
write "\\ USD xi(5,6):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then from the 6x1 entry in  USD J**2 USD  one gets,"$
%USD  J**2^6_1=xi(6,5)*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);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)$
write "\\ USD xi(5,1):=", ws,"USD"$

write "and from the 6x2 entry in  USD J**2 USD  one gets,"$
%USD  J**2^6_2=xi(6,5)*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);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)$
write "\\ USD xi(5,2):=", ws,"USD"$

write "and from the 6x3 entry in  USD J**2 USD  one gets,"$
%USD  J**2^6_3=xi(6,5)*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);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)$
write "\\ USD xi(5,3):=", ws,"USD"$


write "and from the 6x4 entry in  USD J**2 USD  one gets,"$
%USD  J**2^6_4=xi(6,5)*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);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)$
write "\\ USD xi(5,4):=", ws,"USD"$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "then, one gets from  equation USD12|5USD :"$
%{{{1,2},5},
%(xi(6,5)*xi(5,5)*xi(2,2) + xi(6,5)*xi(5,5)*xi(1,1) - xi(6,5)*xi(2,2)*xi(1,1) +
%xi(6,5)*xi(2,1)*xi(1,2) + xi(6,5) - xi(5,5)**2*xi(4,2) + xi(5,5)**2*xi(3,1) - xi
%(4,2) + xi(3,1))/xi(6,5)},
xi(3,1):=-
(xi(6,5)*xi(5,5)*xi(2,2) + xi(6,5)*xi(5,5)*xi(1,1) - xi(6,5)*xi(2,2)*xi(1,1) +
xi(6,5)*xi(2,1)*xi(1,2) + xi(6,5) - xi(5,5)**2*xi(4,2)
- xi(4,2))/    (1+ xi(5,5)**2)$
write "\\ USD xi(3,1):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "then, one gets from  equation USD13|5USD :"$
%{{{1,3},5},
%(xi(6,5)*xi(5,5)*xi(2,3) - xi(6,5)*xi(2,3)*xi(1,1) + xi(6,5)*xi(2,1)*xi(1,3) -
%xi(5,5)**2*xi(4,3)- xi(5,5)**2*xi(2,1)- xi(4,3) - xi(2,1))/xi(6,5)},
xi(4,3):=
(xi(6,5)*xi(5,5)*xi(2,3) - xi(6,5)*xi(2,3)*xi(1,1) + xi(6,5)*xi(2,1)*xi(1,3)
- xi(5,5)**2*xi(2,1)
- xi(2,1) )/(xi(5,5)**2+1)$
write "\\ USD xi(4,3):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "then, one gets from  equation USD14|5USD :"$
%{{{1,4},5},
%(xi(6,5)*xi(5,5)*xi(2,4) - xi(6,5)*xi(2,4)*xi(1,1) + xi(6,5)*xi(2,1)*xi(1,4) -
%xi(5,5)**2*xi(4,4) - xi(5,5)**2*xi(1,1) - xi(4,4) - xi(1,1))/xi(6,5)},
xi(4,4):=
(xi(6,5)*xi(5,5)*xi(2,4) - xi(6,5)*xi(2,4)*xi(1,1) + xi(6,5)*xi(2,1)*xi(1,4)
- xi(5,5)**2*xi(1,1)  - xi(1,1))/(xi(5,5)**2+1)$
write "\\ USD xi(4,4):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "then, one gets from  equation USD14|5USD :"$
%{{{2,3},5},
%( - xi(6,5)*xi(5,5)*xi(1,3) - xi(6,5)*xi(2,3)*xi(1,2) + xi(6,5)*xi(2,2)*xi(1,3)
%- xi(5,5)**2*xi(3,3) - xi(5,5)**2*xi(2,2) - xi(3,3) - xi(2,2))/xi(6,5)},
xi(3,3):=
(- xi(6,5)*xi(5,5)*xi(1,3) - xi(6,5)*xi(2,3)*xi(1,2) + xi(6,5)*xi(2,2)*xi(1,3)
- xi(5,5)**2*xi(2,2)
- xi(2,2))/(xi(5,5)**2+1)$
write "\\ USD xi(3,3):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "then, one gets from  equation USD24|5USD :"$
%{{{2,4},5},
%( - xi(6,5)*xi(5,5)*xi(1,4) - xi(6,5)*xi(2,4)*xi(1,2) + xi(6,5)*xi(2,2)*xi(1,4)
%- xi(5,5)**2*xi(3,4) - xi(5,5)**2*xi(1,2) - xi(3,4) - xi(1,2))/xi(6,5)},
xi(3,4):=
( - xi(6,5)*xi(5,5)*xi(1,4) - xi(6,5)*xi(2,4)*xi(1,2) + xi(6,5)*xi(2,2)*xi(1,4)
- xi(5,5)**2*xi(1,2)
- xi(1,2))/(xi(5,5)**2+1)$
write "\\ USD xi(3,4):=", ws,"\\USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%USD  {J**2}^2_2=xi(2,2)**2 + xi(2,1)*xi(1,2) + xi(3,2)*xi(2,3) + xi(4,2)*xi(2,4)
%;USD\\$
write "At this stage, the 2x2 entry in  USD J**2 USD  is :"$
write "USD  {J**2}^2_2=xi(2,2)**2 + xi(2,1)*xi(1,2) + xi(3,2)*xi(2,3) + xi(4,2)*xi(2,4)
USD"$
write "Hence USD xi(2,1)^2+xi(2,3)^2+xi^2,4)^2 \neq 0 USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%write "Then, the 2x2 entry in  USD J**2 USD  yields :"$
%xi(1,2):=(-1-(xi(2,2)**2 + xi(3,2)*xi(2,3) + xi(4,2)*xi(2,4)))/xi(2,1)$
%write "\\ USD xi(1,2):=", ws,"\\USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$write "Suppose USD xi(2,3) \neq 0 USD"$
write "then, one gets from  equation USD34|5USD :"$
%{{{3,4},5},
%( - xi(6,5)*xi(2,4)*xi(1,3) + xi(6,5)*xi(2,3)*xi(1,4) + xi(5,5)**2*xi(2,4) - xi(
%5,5)**2*xi(1,3) + xi(2,4) - xi(1,3))/xi(6,5)},
xi(1,4):=-
( - xi(6,5)*xi(2,4)*xi(1,3)
%+ xi(6,5)*xi(2,3)*xi(1,4)
+ xi(5,5)**2*xi(2,4) - xi(
5,5)**2*xi(1,3) + xi(2,4) - xi(1,3))/(xi(6,5)*xi(2,3))$
write "\\ USD xi(1,4):=", ws,"\\USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "then, one gets from  equation USD13|6USD :"$
%{{{1,3},6},
%( - xi(6,5)*xi(5,5)*xi(2,3)*xi(2,2) - xi(6,5)*xi(5,5)*xi(2,3)*xi(1,1) + xi(6,5)*
%xi(2,3)*xi(2,2)*xi(1,1) + xi(6,5)*xi(2,3)*xi(1,1)**2 - xi(6,5)*xi(2,2)*xi(2,1)*
%xi(1,3) - xi(6,5)*xi(2,1)*xi(1,3)*xi(1,1) + xi(5,5)**2*xi(4,2)*xi(2,3) + xi(5,5)
%**2*xi(4,1)*xi(1,3) + xi(5,5)**2*xi(2,2)*xi(2,1) + xi(5,5)**2*xi(2,1)*xi(1,1) +
%xi(4,2)*xi(2,3) + xi(4,1)*xi(1,3) + xi(2,2)*xi(2,1) + xi(2,1)*xi(1,1))/(xi(5,5)
%**2 + 1)},

xi(4,2):=-
( - xi(6,5)*xi(5,5)*xi(2,3)*xi(2,2) - xi(6,5)*xi(5,5)*xi(2,3)*xi(1,1) + xi(6,5)*
xi(2,3)*xi(2,2)*xi(1,1) + xi(6,5)*xi(2,3)*xi(1,1)**2 - xi(6,5)*xi(2,2)*xi(2,1)*
xi(1,3) - xi(6,5)*xi(2,1)*xi(1,3)*xi(1,1)
%+ xi(5,5)**2*xi(4,2)*xi(2,3)
+ xi(5,5)**2*xi(4,1)*xi(1,3) + xi(5,5)**2*xi(2,2)*xi(2,1) + xi(5,5)**2*xi(2,1)*xi(1,1)
%+  xi(4,2)*xi(2,3)
+ xi(4,1)*xi(1,3) + xi(2,2)*xi(2,1) + xi(2,1)*xi(1,1))/((xi(5,5)**2 + 1)*xi(2,3))$
write "\\ USD xi(4,2):=", ws,"\\USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Trace J:=(xi(6,5)*xi(5,5)*xi(2,4)*xi(2,3) - xi(6,5)*xi(5,5)*xi(2,3)*xi(1,3) - xi
%(6,5)*xi(2,4)*xi(2,3)*xi(1,1) + xi(6,5)*xi(2,4)*xi(2,1)*xi(1,3) - xi(6,5)*xi(2,3
%)**2*xi(1,2) + xi(6,5)*xi(2,3)*xi(2,2)*xi(1,3) - xi(5,5)**2*xi(2,4)*xi(2,1) + xi
%(5,5)**2*xi(2,1)*xi(1,3) - xi(2,4)*xi(2,1) + xi(2,1)*xi(1,3))/(xi(2,3)*(xi(5,5)
%**2 + 1))$

xi(1,2):=(xi(6,5)*xi(5,5)*xi(2,4)*xi(2,3) - xi(6,5)*xi(5,5)*xi(2,3)*xi(1,3) - xi
(6,5)*xi(2,4)*xi(2,3)*xi(1,1) + xi(6,5)*xi(2,4)*xi(2,1)*xi(1,3)
%- xi(6,5)*xi(2,3)**2*xi(1,2)
+ xi(6,5)*xi(2,3)*xi(2,2)*xi(1,3) - xi(5,5)**2*xi(2,4)*xi(2,1) + xi
(5,5)**2*xi(2,1)*xi(1,3) - xi(2,4)*xi(2,1) + xi(2,1)*xi(1,3))/
(xi(2,3)**2*xi(6,5))$
write "\\ USD xi(1,2):=", ws,"\\USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "then, one gets from  equation USD23|6USD :"$
%{{{2,3},6},
xi(3,2):=-
 (xi(6,5)**2*xi(5,5)*xi(2,4)*xi(2,3)**2*xi(2,2) + xi(6,5)**2*xi
(5,5)*xi(2,4)*xi(2,3)**2*xi(1,1) - xi(6,5)**2*xi(5,5)*xi(2,4)*xi(2,3)*xi(2,1)*xi
(1,3) + xi(6,5)**2*xi(5,5)*xi(2,3)*xi(2,1)*xi(1,3)**2 - xi(6,5)**2*xi(2,4)*xi(2,
3)**2*xi(2,2)*xi(1,1) - xi(6,5)**2*xi(2,4)*xi(2,3)**2*xi(1,1)**2 + xi(6,5)**2*xi
(2,4)*xi(2,3)*xi(2,2)*xi(2,1)*xi(1,3) + 2*xi(6,5)**2*xi(2,4)*xi(2,3)*xi(2,1)*xi(
1,3)*xi(1,1) - xi(6,5)**2*xi(2,4)*xi(2,1)**2*xi(1,3)**2 - xi(6,5)**2*xi(2,3)**2*
xi(1,3)*xi(1,1)**2 - xi(6,5)**2*xi(2,3)**2*xi(1,3) + xi(6,5)**2*xi(2,3)*xi(2,1)*
xi(1,3)**2*xi(1,1) + xi(6,5)*xi(5,5)**3*xi(2,4)*xi(2,3)*xi(2,1) - xi(6,5)*xi(5,5
)**3*xi(2,3)*xi(2,1)*xi(1,3) - xi(6,5)*xi(5,5)**2*xi(4,1)*xi(2,3)*xi(1,3)**2 -
xi(6,5)*xi(5,5)**2*xi(2,4)*xi(2,3)*xi(2,2)*xi(2,1) - 2*xi(6,5)*xi(5,5)**2*xi(2,4
)*xi(2,3)*xi(2,1)*xi(1,1) + 2*xi(6,5)*xi(5,5)**2*xi(2,4)*xi(2,1)**2*xi(1,3) + xi
(6,5)*xi(5,5)**2*xi(2,3)**2*xi(2,2)**2 + xi(6,5)*xi(5,5)**2*xi(2,3)**2 + xi(6,5)
*xi(5,5)**2*xi(2,3)*xi(2,2)*xi(2,1)*xi(1,3) - xi(6,5)*xi(5,5)**2*xi(2,1)**2*xi(1
,3)**2 + xi(6,5)*xi(5,5)*xi(2,4)*xi(2,3)*xi(2,1) - xi(6,5)*xi(5,5)*xi(2,3)*xi(2,
1)*xi(1,3) - xi(6,5)*xi(4,1)*xi(2,3)*xi(1,3)**2 - xi(6,5)*xi(2,4)*xi(2,3)*xi(2,2
)*xi(2,1) - 2*xi(6,5)*xi(2,4)*xi(2,3)*xi(2,1)*xi(1,1) + 2*xi(6,5)*xi(2,4)*xi(2,1
)**2*xi(1,3) + xi(6,5)*xi(2,3)**2*xi(2,2)**2 + xi(6,5)*xi(2,3)**2 + xi(6,5)*xi(2
,3)*xi(2,2)*xi(2,1)*xi(1,3) - xi(6,5)*xi(2,1)**2*xi(1,3)**2 - xi(5,5)**4*xi(2,4)
*xi(2,1)**2 + xi(5,5)**4*xi(2,1)**2*xi(1,3) - 2*xi(5,5)**2*xi(2,4)*xi(2,1)**2 +
2*xi(5,5)**2*xi(2,1)**2*xi(1,3) - xi(2,4)*xi(2,1)**2 + xi(2,1)**2*xi(1,3))/(xi(6
,5)*xi(2,3)**2*(xi(5,5)**2 + 1)*xi(2,3))$
write "\\ USD xi(3,2):=", ws,"\\USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "If we suppose USD xi(2,4) =xi(2,3) USD"$
%xi(2,4):=xi(1,3)$
%write "\\ USD xi(2,4):=", ws,"\\USD"$
%{{{1,4},6},
%(xi(2,3)**2*xi(1,1)**2 + xi(2,3)**2 - 2*xi(2,3)*xi(2,1)*xi(1,3)*xi(1,1) + xi(2,1
%)**2*xi(1,3)**2)/xi(2,3)**2},
write "Then we get that equation USD14|6 USD reads"$
write "USD (xi(2,3)xi(1,1)-xi(2,1)xi(1,3))**2  +xi(2,3)**2=0USD"$
write "Hence this case is impossible."$
write "that i : USD xi(2,4) \neq xi(2,3) USD"$


write "Then, the 2x1 entry in  USD J**2 USD  yields :"$
%USD  {J**2}^2_1=xi(4,1)*(xi(2,4) - xi(1,3)) + ( - xi(6,5)*xi(5,5)*xi(2,4)*xi(2,3
%)*xi(2,1) + xi(6,5)*xi(5,5)*xi(2,3)*xi(2,1)*xi(1,3) + xi(6,5)*xi(2,4)*xi(2,3)*xi
%(2,1)*xi(1,1) - xi(6,5)*xi(2,4)*xi(2,1)**2*xi(1,3) - xi(6,5)*xi(2,3)**2*xi(1,1)
%**2 - xi(6,5)*xi(2,3)**2 + xi(6,5)*xi(2,3)*xi(2,1)*xi(1,3)*xi(1,1) + xi(5,5)**2*
%xi(2,4)*xi(2,1)**2 - xi(5,5)**2*xi(2,1)**2*xi(1,3) + xi(2,4)*xi(2,1)**2 - xi(2,1
%)**2*xi(1,3))/(xi(2,3)*(xi(5,5)**2 + 1));USD\\$

xi(4,1):=-
 ( - xi(6,5)*xi(5,5)*xi(2,4)*xi(2,3
)*xi(2,1) + xi(6,5)*xi(5,5)*xi(2,3)*xi(2,1)*xi(1,3) + xi(6,5)*xi(2,4)*xi(2,3)*xi
(2,1)*xi(1,1) - xi(6,5)*xi(2,4)*xi(2,1)**2*xi(1,3) - xi(6,5)*xi(2,3)**2*xi(1,1)
**2 - xi(6,5)*xi(2,3)**2 + xi(6,5)*xi(2,3)*xi(2,1)*xi(1,3)*xi(1,1) + xi(5,5)**2*
xi(2,4)*xi(2,1)**2 - xi(5,5)**2*xi(2,1)**2*xi(1,3) + xi(2,4)*xi(2,1)**2 - xi(2,1
)**2*xi(1,3))/(xi(2,3)*(xi(5,5)**2 + 1)*(xi(2,4)-xi(1,3)))$
write "\\ USD xi(4,1):=", ws,"\\USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%off exp$
%on factor$

WRITE " \par Now the nonzero torsion equations left are :"$

%COMMENT
write COLLECT_EQ$

%%%%%%%
%La matrice MATJ de J :
write "\par The matrix USD J USD is :\\"$

 MATRIX MATJ(DIM,DIM)$
 FOR i:=1:DIM DO FOR j:=1:DIM DO MATJ(i,j):=xi(i,j)$

 %WRITE "Matrice de J:=",MATJ$

 FOR i:=1:DIM DO FOR j:=1:DIM DO
 <<WRITE "USD  J^",i,"_",j,"=",MATJ(i,j),";USD\\">>$

 MATJCARRE:=(MATJ)**2$

 %WRITE "Matrice de J**2:=",(MATJ)**2$

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  FOR i:=1:DIM DO FOR j:=1:DIM DO
  <<WRITE "USD  {J**2}^",i,"_",j,"=",MATJCARRE(i,j),";USD\\">>$

 WRITE "\\$ det J:=",DET(MATJ)$
 WRITE "Trace J:=",TRACE(MATJ)$

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 out "matJ6_6.red"$

WRITE "% Writing the entries of J:"$
WRITE "% J(i,k) denotes the entry in the ith row , kth column"$
WRITE "% i.e. stands for  the  LaTex expression J^i_k"$
WRITE "% Similarly, xi(i,k) stands for the  LaTex expression \xi^i_k"$
write "matrix J(6,6)"$
 FOR i:=1:DIM DO FOR j:=1:DIM DO
 <<WRITE "J(",i,",",j,"):=",MATJ(i,j)>>$


 %FOR i:=1:DIM DO FOR j:=1:DIM DO
 %<<WRITE MATJ(i,j):=MATJ(i,j)>>$

 WRITE "%where the parameters are subject to the following condition"$
 write "%USDxi(2,3)*xi(6,5)*(xi(1,3)-xi(2,4)) \neq 0. USD"$

 write "END"$
 shut "matJ6_6.red"$
 out "rprogram6_6.tex"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 WRITE "\end{document}"$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 bye;