%off echo,nat$
off echo$
out "rreducparautommodg6_g4xC2case2N5.r"$
write "rreducparautommodg6_g4xC2case2N5.r"$
operator b$
ON REVPRI$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "The generic automorphism phi of  g_{4} x C**2 as computed by calculautom6_g4xC2.red :"$

phi:=
mat((b(1,1),0,0,0,0,0),(b(2,1),b(2,2),0,0,0,0),(b(3,1),b(3,2),b(2,2)*b(1,1),0,0,
0),(b(4,1),b(4,2),b(3,2)*b(1,1),b(2,2)*b(1,1)**2,b(4,5),b(4,6)),(b(5,1),b(5,2),0
,0,b(5,5),b(5,6)),(b(6,1),b(6,2),0,0,b(6,5),b(6,6)))$




write "phi:=",phi;
on factor$
write "det(phi):=",det(phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generic derivation  :
operator xi$

delta:=
mat((xi(1,1),0,0,0,0,0),
(xi(2,1),xi(2,2),0,0,0,0),
(xi(3,1),xi(3,2),xi(1,1)+xi(2,2),0,0,0),
(xi(4,1),xi(4,2),xi(3,2),2*xi(1,1)+xi(2,2),xi(4,5),xi(4,6)),
(xi(5,1),xi(5,2),0,0,xi(5,5),xi(5,6)),
(xi(6,1),xi(6,2),0,0,xi(6,5),xi(6,6)))$

write "generic derivation : delta:=",delta;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The nonzero adjoint derivations
matrix adx1(6,6)$
adx1:=
sub({xi(1,1)=0,xi(2,1)=0,xi(2,2)=0,xi(3,1)=0,xi(3,2)=1,xi(4,1)=0,xi(4,2)=0,xi(4,5)=0,xi(4,6)=0,xi(5,1)=0,xi(5,2)=0,xi(5,5)=0,xi(5,6)=0,xi(6,1)=0,xi(6,2)=0,xi(6,5)=0,xi(6,6)=0}, delta)$
matrix adx2(6,6)$
adx2:=
sub({xi(1,1)=0,xi(2,1)=0,xi(2,2)=0,xi(3,1)=-1,xi(3,2)=0,xi(4,1)=0,xi(4,2)=0,xi(4,5)=0,xi(4,6)=0,xi(5,1)=0,xi(5,2)=0,xi(5,5)=0,xi(5,6)=0,xi(6,1)=0,xi(6,2)=0,xi(6,5)=0,xi(6,6)=0}, delta)$
matrix adx3(6,6)$
adx3:=
sub({xi(1,1)=0,xi(2,1)=0,xi(2,2)=0,xi(3,1)=0,xi(3,2)=0,xi(4,1)=-1,xi(4,2)=0,xi(4,5)=0,xi(4,6)=0,xi(5,1)=0,xi(5,2)=0,xi(5,5)=0,xi(5,6)=0,xi(6,1)=0,xi(6,2)=0,xi(6,5)=0,xi(6,6)=0}, delta)$
%matrix adx4(6,6)$
%adx4:=
%matrix adx5(6,6)$
%adx5:=
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
on nat$
write adx1:=adx1$
write adx2:=adx2$
write adx3:=adx3$
%write adx4:=adx4$
%write adx5:=adx5$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "The generic nilpotent derivation : the eigenvalues are 0"$
xi(1,1):=0$
write "xi(1,1):=",xi(1,1)$
xi(2,2):=0$
write "xi(2,2):=",xi(2,2)$

write "And the matrix A:=(xi(5,5),xi(5,6)),(xi(6,5),xi(6,6)) is nilpotent"$
write "We hence get 2 cases according to whether A neq 0 or A=0."$
write "We consider here the case 2 where A := 0."$
xi(5,5):=0$
xi(5,6):=0$
xi(6,5):=0$
xi(6,6):=0$
for i:=5:6 do for j:=5:6 do <<write "xi(",i,",",j,"):=",xi(i,j)>>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% by subtracting adjoints one then may suppose
xi(3,1):=0$
xi(3,2):=0$
xi(4,1):=0$
write "by subtracting adjoints one then may suppose:"$
write "xi(3,1):=0,xi(3,2):=0,xi(4,1):=0"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "delta:=",delta;
write "We denote this delta by the shortform"$
shortformdelta:=
{xi(2,1),SS,xi(4,2),xi(4,5),xi(4,6),SS,xi(5,1),xi(5,2),SS,xi(6,1),xi(6,2)}$
paramindexeslist:=
{{2,1},{4,2},{4,5},{4,6},{5,1},{5,2},{6,1},{6,2}}$

write "shortformdelta:=", shortformdelta$
write "paramindexeslist:=",paramindexeslist$

off nat$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE SHORTFORM(M0)$
BEGIN$
M:=M0$
WS:=
{M(2,1),SS,M(4,2),M(4,5),M(4,6),SS,M(5,1),M(5,2),SS,M(6,1),M(6,2)}$
RETURN WS$
END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE DELTAPRIMEMODADG(M0,AUTOM)$
BEGIN $
M:=M0$
M:=AUTOM*M*AUTOM**(-1)$
M:=M-M(3,2)*adx1 +M(3,1)*adx2 + M(4,1)*adx3 $

IF AUTOM=phi THEN  <<write "With the generic automorphism one gets">>$
%IF AUTOM=phi THEN  <<write "With the first kind automorphism one gets">>$
IF AUTOM=psi THEN  <<write "With the second kind automorphism one gets">>$

write "shortformdeltaprimemodadg:=",shortform(M)$
for each U in paramindexeslist do <<write "deltaprimemodg(" ,part(U,1), ",", part(U,2), "):=" ,
                  M(part(U,1),part(U,2))>>$
on factor$
write "det(AUTOM):=",det(AUTOM);
DELTAPRIMEMODADG:=M$
on nat$
write "DELTAPRIMEMODADG:=", M$
off nat$
CLEAR M$
%RETURN DELTAPRIMEMODADG$
END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
comment
deltaprimemodg(4,5):=(b(2,2)*b(1,1)**2*( - b(6,5)*xi(4,6) + b(6,6)*xi(4,5)))/( -
 b(6,5)*b(5,6) + b(6,6)*b(5,5))$
comment
deltaprimemodg(4,6):=(b(2,2)*b(1,1)**2*(b(5,5)*xi(4,6) - b(5,6)*xi(4,5)))/( - b(
6,5)*b(5,6) + b(6,6)*b(5,5))$
%det(AUTOM):=( - b(6,5)*b(5,6) + b(6,6)*b(5,5))*b(2,2)**3*b(1,1)**4$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "****************** SUBCASE 2 : xi(4,5),xi(4,6)  BOTH 0 *********************"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
xi(4,5):=0$
write "xi(4,5):=",xi(4,5)$
xi(4,6):=0$
write "xi(4,6):=",xi(4,6)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "*************** SUBCASE 2.2 : xi(5,2),xi(6,2)  BOTH 0 *********************"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
xi(5,2):=0$
write "xi(5,2):=",xi(5,2)$
xi(6,2):=0$
write "xi(6,2):=",xi(6,2)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "**** Suppose xi(2,1) neq 0."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "******* Then if xi(2,1) neq 0, we get"$
write "deltaprime(2,1)=k, and"$
write "deltaprime(5,1)=0, and"$
write "deltaprime(6,1)=0 by taking:"$
b(2,2):=k*b(1,1)/xi(2,1)$
%deltaprimemodg(5,1):=(b(5,2)*xi(2,1) + b(5,5)*xi(5,1) + b(5,6)*xi(6,1))/b(1,1)$
b(5,2):=-( b(5,5)*xi(5,1) + b(5,6)*xi(6,1))/xi(2,1)$
%deltaprimemodg(6,1):=(b(6,2)*xi(2,1) + b(6,5)*xi(5,1) + b(6,6)*xi(6,1))/b(1,1)$
b(6,2):=-( b(6,5)*xi(5,1) + b(6,6)*xi(6,1))/xi(2,1)$
write "b(2,2):=",b(2,2)$
write "b(5,2):=",b(5,2)$
write "b(6,2):=",b(6,2)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence in that case C x_5 +C x_6 would be a direct factor of gtildedelta."$
write "Hence we dismiss that case."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "**** Suppose xi(2,1) = 0."$
xi(2,1):=0$
write "xi(2,1):=",xi(2,1)$
clear b(2,2),b(5,2),b(6,2)$
write "clear b(2,2),b(5,2),b(6,2)"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "In that case xi(5,1),xi(6,1) cannot be both zero, since then"$
write "C x_5 +C x_6 would again be a direct factor of gtildedelta."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "*************** Hence we suppose: xi(5,1),xi(6,1) NOT  BOTH 0 *****************"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "In that subcase, one may choose the invertible matrix"$
write "((xi(5,5),xi(5,6)),(xi(6,5),xi(6,6)) such that"$
write "deltaprime(5,1):=0, deltaprime(6,1):=1."$
write "Hence we may suppose:"$
xi(5,1):=0$
xi(6,1):=1$
write "xi(5,1):=",xi(5,1)$
write "xi(6,1):=",xi(6,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "We keep deltaprime(5,1):=0 and deltaprime(6,1):=k by taking:"$

b(5,6):=0$
b(6,6):=k*b(1,1)$
write "b(5,6):=",b(5,6)$
write "b(6,6):=",b(6,6)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
DELTAPRIMEMODADG(delta,phi);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Hence in that case C x_5  would be a direct factor of gtildedelta."$
write "Hence we dismiss that case as well."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
bye$