%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%This is "2009structcompln2case2global.red"    :
% renamed:
% "n2case3.red"
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%  PRECISER SORTIE RESULTATS   %%%%%%%%%%%%%%%%%%%%%
%ON FACTOR$
   off echo$
    off nat$
OUT "rn2case3.r"    $
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 ON RAT$
 OFF MSG$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%% Loading the commutation relations file  %%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%DIM:=(dimension de l'algebre)$
DIM:= 6$
  in "n.2"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%ECRITURE DES RELATIONS DE COMMUTATION EN TEX
WRITE "\documentclass{article}"$
WRITE "\usepackage{amsmath,amssymb}"$
WRITE "\sloppy"$
WRITE "\begin{document}"$
%WRITE "This output from the file \texttt{structcompln2\_2.red}.\\"$
WRITE "Computation of all complex    structures on the real Lie Algebra"$
write "USD {\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}.USD"$

write "\\Case 3."$

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"$
%FOR j:=1:DIM DO <<CLEAR X(j)>>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE OTEZERO(U)$  %enleve les 0 dans une liste
BEGIN$
LIST U$
RETURN FOR EACH A IN U JOIN IF A NEQ 0 THEN {A} ELSE {}$
END$

PROCEDURE CALLLET(U,V)$
LET U=V$

PROCEDURE UNKNOWNS(U)$
BEGIN$ LIST U$
RETURN  IF U={} THEN U ELSE
        IF PART(PART(SOLVE(U),1),0) = LIST
          THEN FOR EACH A IN PART(SOLVE(U),1) COLLECT LHS A
        ELSE {LHS PART(SOLVE(U),1)} $
END$


PROCEDURE SSOLVE(n,p)$
BEGIN INTEGER k,j$
 k:=1$ j:=1$
S1: IF z(k,j) NEQ 0 THEN <<GO TO S2>> ELSE <<GO TO S3>>$
S2: CALLLET(z(k,j),0) $
S3: IF j<p THEN j:=j+1 ELSE IF k<n  THEN <<k:=k+1$ j:=1>> ELSE GO TO
 S4$
GO TO S1$
S4: END$

PROCEDURE distinct(U)$
BEGIN;INTEGER j;LIST UU$ j:=1;UU:={}$
S: IF U NEQ {} THEN    ZZZ:=PART(u,j)$
IF ZZZ MEMBER UU THEN <<GO TO P>> ELSE UU:= ZZZ. UU $
P: IF j<LENGTH(U) THEN <<j:=j+1,GO TO S>> ELSE <<RETURN REVERSE(UU)>>$
CLEAR ZZZ$
END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%collecting the torsion equations%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 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)$
 %LIGNE A MODIFIER POUR LA LINEARITE SUR LE PARAMETRE CONTINU L
 for all j let f(L*x(j))=L*f(x(j))$
 for all j let f(-L*x(j))=-L*f(x(j))$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%FOR j:=1:DIM COLLECT X(j)$
%FOR EACH W IN WS DO <<FACTOR W>>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 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"$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE LOCALRECAP$
BEGIN$
write "localrecap"$
WRITE "USD \par Now the nonzero torsion equations left are :"$

write COLLECT_EQ$

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)$

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

 MATJCARRE:=(MATJ)**2$

for i:=1:6 do for j:=1:6 do << write "\\USD J^2(", i ,",", j, "):=" , MATJCARRE(i,j),"USD\\">>$
% FOR i:=1:DIM DO FOR j:=1:DIM DO
% <<WRITE "USD  J^2(",i,",",j,"):=",MATJCARRE(i,j),";USD\\">>$

write "Trace(J):=", trace(matJ)$
on nat$ write "J:=",matJ; off nat$
on nat$ write "J**2:=",matJcarre; off nat$


on factor$
write "det(J):=", det(matJ)$
off factor$
END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%Computing the complex structures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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 USD15|2USD :"$
xi(2,5):=0$
write "\\ USD xi(2,5):=", ws,"USD"$
write "\\ and from equation USD25|1USD :"$
xi(1,5):=0$
write "\\ USD xi(1,5):=", ws,"USD"$
write "\\ and from equation USD36|4USD :"$
xi(4,6):=0$
write "\\ USD xi(4,6):=", ws,"USD"$
write "\\ and from equation USD46|3USD :"$
xi(3,6):=0$
write "\\ USD xi(3,6):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "*******************************************************************"$
write "Here in case 3, we suppose that :"$
xi(1,1):=0$
xi(2,1):=1$
xi(5,1):=0$
xi(5,2):=0$
write "\\ USD xi(1,1):=", xi(1,1),"USD"$
write "\\ USD xi(2,1):=", xi(2,1),"USD"$
write "\\ USD xi(5,1):=", xi(5,1),"USD"$
write "\\ USD xi(5,2):=", xi(5,2),"USD"$


xi(4,4):=0$
xi(3,4):=-xi(3,3)$
xi(6,3):=0$
xi(6,4):=0$
write "\\ USD xi(4,4):=", xi(4,4),"USD"$
write "\\ USD xi(3,4):=", xi(3,4),"USD"$
write "\\ USD xi(6,4):=", xi(6,4),"USD"$
write "\\ USD xi(6,3):=", xi(6,3),"USD"$
write "*******************************************************************"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%localrecap()$
%%%%%%%%%%%%%%%%%%%%é%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then from equation"$
%{{{1,2},3},xi(3,5)*xi(2,2)},
write "%{{{1,2},3},xi(3,5)*xi(2,2)},"$
xi(3,5):=0$
write "\\ USD xi(3,5):=", xi(3,5),"USD"$
write "%{{{1,2},4},xi(4,5)*xi(2,2)},"$
%{{{1,2},4},xi(4,5)*xi(2,2)},
xi(4,5):=0$
write "\\ USD xi(4,5):=", xi(4,5),"USD"$
%{{{3,4},1},xi(3,3)*xi(1,6)},
write "%{{{3,4},1},xi(3,3)*xi(1,6)},"$
xi(1,6):=0$
write "\\ USD xi(1,6):=", xi(1,6),"USD"$
%{{{3,4},2},xi(3,3)*xi(2,6)},
write "%{{{3,4},2},xi(3,3)*xi(2,6)}, "$
xi(2,6):=0$
write "\\ USD xi(2,6):=", xi(2,6),"USD"$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%localrecap()$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "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\\$
write "%\\USD J^2(6,5):=(xi(6,6) + xi(5,5))*xi(6,5)USD\\"$
write "%\\USD J^2(6,6):=xi(6,6)**2 + xi(6,5)*xi(5,6)USD\\"$
xi(6,6):=-xi(5,5)$
write "\\ USD xi(6,6):=", xi(6,6),"USD"$
write "and from the trace"$
%Trace(J):=xi(3,3) + xi(2,2) + xi(5,5) + xi(6,6)$
write "%Trace(J):=xi(3,3) + xi(2,2) + xi(5,5) + xi(6,6)$"$
xi(2,2):=-xi(3,3)$
write "\\ USD xi(2,2):=", xi(2,2),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%localrecap()$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "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))},
write "%{{{1,2},5},xi(1,2) + 1 - xi(5,5)*xi(3,3)},   "$
write "%{{{3,4},6}, - (xi(4,3)*xi(3,3) - 1 + xi(5,5)*xi(3,3))},  "$
xi(1,2):= - xi(4,3)*xi(3,3)$
write "\\ USD xi(1,2):=", xi(1,2),"USD"$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%localrecap()$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then from the equation"$
%\\USD J^2(6,1):=xi(6,2) - xi(6,1)*xi(5,5)USD\\$
xi(6,2) :=xi(6,1)*xi(5,5)$
write "%\\USD J^2(6,1):=xi(6,2) - xi(6,1)*xi(5,5)USD\\$ "$
write "\\ USD xi(6,2):=", xi(6,2),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%localrecap()$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then from the equations"$

%{{{3,4},6}, - xi(5,5)*xi(3,3) - xi(4,3)*xi(3,3) + 1},
write "%{{{3,4},6}, - xi(5,5)*xi(3,3) - xi(4,3)*xi(3,3) + 1},"$
%\\USD J^2(6,2):= - xi(6,1)*(xi(5,5)**2 + xi(5,5)*xi(3,3) + xi(4,3)*xi(3,3))USD\\$
write "%\\USD J^2(6,2):= - xi(6,1)*(xi(5,5)**2 + xi(5,5)*xi(3,3) + xi(4,3)*xi(3,3))USD\\$ "$
xi(6,1):=0$
write "\\ USD xi(6,1):=", xi(6,1),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%localrecap()$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then from the equations"$
%\\USD J^2(1,4):= - xi(3,3)*(xi(4,3)*xi(2,4) + xi(1,3))USD\\$
xi(1,3):=-xi(4,3)*xi(2,4) $
write "%\\USD J^2(1,4):= - xi(3,3)*(xi(4,3)*xi(2,4) + xi(1,3))USD\\$   "$
write "\\ USD xi(1,3):=", xi(1,3),"USD"$


%\\USD J^2(4,1):=xi(4,3)*xi(3,1) + xi(4,2)USD\\$
write "%\\USD J^2(4,1):=xi(4,3)*xi(3,1) + xi(4,2)USD\\$  "$
xi(4,2):=-xi(4,3)*xi(3,1) $
write "\\ USD xi(4,2):=", xi(4,2),"USD"$

%\\USD J^2(6,3):=xi(6,5)*xi(5,3)USD\\$
write "%\\USD J^2(6,3):=xi(6,5)*xi(5,3)USD\\$ "$

%\\USD J^2(6,4):=xi(6,5)*xi(5,4)USD\\$
write "%\\USD J^2(6,4):=xi(6,5)*xi(5,4)USD\\$ "$
xi(5,3):=0$
xi(5,4):=0$
write "\\ USD xi(5,3):=", xi(5,3),"USD"$
write "\\ USD xi(5,4):=", xi(5,4),"USD"$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%localrecap()$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then from the equations"$

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
localrecap()$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%{{{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)},
%As xi(6,5) neq 0, necessarily
%xi(4,3)*xi(3,3)*xi(3,1)**2 + xi(3,3)*xi(3,2)*xi(3,1) + xi(3,2)**2  neq 0
% Hence, if xi(3,1) neq 0,  with a first kind automorphism having diag(U,I)
% with U=b*I+J_1 where J_1 =mat((0,-xi(4,3)*xi(3,3)),(1,-xi(3,3))) and b=-xi(3,2)/xi(3,1)
%J_3 = mat((xi(3,1),xi(3,2)),(xi(4,1),-xi(4,3)*xi(3,1))) is changed to
% J_3*U =mat((0, *),(*,*))
% U= (mat(-xi(3,2)/xi(3,1), xi(xi(4,3)*xi(3,3)),(1,-xi(3,2)/xi(3,1)-xi(3,3))
on nat$
matJ_1 := mat((0,-xi(4,3)*xi(3,3)),(1,-xi(3,3)))$
matJ_3 := mat((xi(3,1),xi(3,2)),(xi(4,1),-xi(4,3)*xi(3,1)))$
write "matJ_1:=",matJ_1$
write "matJ_3:=",matJ_3$
write "matU:= bI+ J_1"$
MatU:=
mat((b,-xi(4,3)*xi(3,3)),(1,b-xi(3,3)))$
write "matU:=",matU$
write "matJ_3*matU:=",matJ_3*matU$
b:=-xi(3,2)/xi(3,1)$
write "matU:=",matU$
write "matJ_3:=",matJ_3$
write "matJ_3*matU:=",matJ_3*matU$
write "det(matU):=",det matU$
off nat$
clear b$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
xi(3,1):=0$   %PAR PASSAGE A UN J EQUIVALENT
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
localrecap()$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


write "Then from the equations"$
%{{{1,4},6},xi(6,5)*xi(2,4) - xi(3,2)},
write "%{{{1,4},6},xi(6,5)*xi(2,4) - xi(3,2)},   "$
xi(6,5):=xi(3,2)/xi(2,4)$
write "\\ USD xi(6,5):=", xi(6,5),"USD"$

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

%{{{1,4},5},   ( - xi(4,3)*xi(3,3)**2 + xi(3,3) + xi(3,2)*xi(1,4))/xi(3,2)},
write "%{{{1,4},5},   ( - xi(4,3)*xi(3,3)**2 + xi(3,3) + xi(3,2)*xi(1,4))/xi(3,2)}, "$
xi(1,4):=-( - xi(4,3)*xi(3,3)**2 + xi(3,3) )/xi(3,2)$
write "\\ USD xi(1,4):=", xi(1,4),"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},
write "%{{{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},  "$
xi(2,3):=-( - xi(4,3)*xi(3,3) + xi(3,3)**2 + 1)/xi(3,2)$
write "\\ USD xi(2,3):=", xi(2,3),"USD"$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
localrecap()$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Finally, with a Phi having diag(bI,I) with suitable b=xi(3,3)/xi(3,2),"$
write "one may suppose xi(3,2)=xi(3,3)"$
xi(3,2):=xi(3,3)$
write "\\ USD xi(3,2):=", xi(3,2),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
localrecap()$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%BYE$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ Then USD J USD has entries :"$
off exp$
on factor$
 FOR i:=1:DIM DO FOR j:=1:DIM DO
 <<WRITE "USD  J^",i,"_",j,"=",MATJ(i,j),";USD\\">>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 write "Hence we are finally reduced to "$

 %%%%%%%%%%%%%%%%%
off exp$
on factor$
write "\\"$
write "{\fontsize{8}{10} \selectfont"$
 write "USDUSD J(\xi^3_3,\xi^4_3) := \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write MATJ(i,j), "&">>
                         else    <<write MATJ(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write MATJ(i,j), "&">>
                         else    <<write MATJ(i,j), "\end{pmatrix}USDUSD" >>$
                         write "}"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "with condition USD xi(3,3) \neq 0 USD."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 WRITE "\par Commutation relations of USD \mathfrak{m} : USD"$
FOR i:=1:DIM DO <<FACTOR x(i)>>$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO X_(i,j):=x(i)*x(j)$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO Y_(i,j):=F(x(i))*F(x(j))$
FOR j:=1:DIM DO X(j):=MKID(tildex_,j)$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO
    IF X_(i,j)-Y_(i,j) NEQ 0 THEN WRITE
        "USD[\tilde{x}_",i,",\tilde{x}_",j,"]=", X_(i,j) - Y_(i,j),"USD;"$
WRITE "\P"$
FOR j:=1:DIM DO <<CLEAR X(j)>>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par Now we check if the condition USD [Jx,y]= J[x,y] USD is satisfied"$
write "USD\forall x,y \in {\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "},USD"$
write "\textit{i.e.} if USD{\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}USD"$
write "is a \textit{complex} algebra."$

for all j,k let B1(j,k) = F(x(j)*x(k))$
for all j,k let B(j,k) = - B1(j,k) + F(x(j))*x(k)$
COLLECT_ALGEBRECOMPLEXE:=FOR j1:=1:DIM JOIN
   FOR j2:=1:DIM JOIN
   IF B(j1,j2) NEQ 0 THEN
   {{{j1,j2},B(j1,j2)}}
   ELSE {}$
IF LENGTH(COLLECT_ALGEBRECOMPLEXE) NEQ 0 THEN WRITE "\\USD J[x_j,x_k] \neq [Jx_j,x_k] USD
in the following cases",COLLECT_ALGEBRECOMPLEXE
   ELSE WRITE "\\USD{\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}USD is a COMPLEX ALGEBRA"$
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write "\end{document}"$
bye$
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