The generic automorphism phi of g_{6,9} as computed by calculautom6_9.red :

They fall into 2 kinds.

First kind :

phi:=

mat((b(1,1),0,0,0,0,0),

    (0,b(2,2),0,0,0,0),

    (0,0,b(3,3),0,0,0),

    (b(4,1),b(4,2),b(4,3),b(2,2)*b(1,1),0,0),

                    b(4,2)*b(3,3)
    (b(5,1),b(5,2),---------------,0,b(3,3)*b(1,1),0),
                       b(2,2)

    (b(6,1),b(6,2),b(6,3), - b(5,1)*b(2,2), - b(4,1)*b(3,3),b(3,3)*b(2,2)*b(1,1)

     ))


                3       3       4
det(phi):=b(3,3) *b(2,2) *b(1,1)

Second kind :

psi:=

mat((bb(1,1),0,0,0,0,0),

    (0,0,bb(2,3),0,0,0),

    (0,bb(3,2),0,0,0,0),

    (bb(4,1),bb(4,2),bb(4,3),0,bb(2,3)*bb(1,1),0),

              bb(4,3)*bb(3,2)
    (bb(5,1),-----------------,bb(5,3),bb(3,2)*bb(1,1),0,0),
                  bb(2,3)

    (bb(6,1),bb(6,2),bb(6,3), - bb(4,1)*bb(3,2), - bb(5,1)*bb(2,3),

     bb(3,2)*bb(2,3)*bb(1,1)))


                 3        3        4
det(psi):=bb(3,2) *bb(2,3) *bb(1,1)

generic derivation : delta:=

mat((xi(1,1),0,0,0,0,0),

    (0,xi(2,2),0,0,0,0),

    (0,0,xi(3,3),0,0,0),

    (xi(4,1),xi(4,2),xi(4,3),xi(1,1) + xi(2,2),0,0),

    (xi(5,1),xi(5,2),xi(4,2),0,xi(1,1) + xi(3,3),0),

    (xi(6,1),xi(6,2),xi(6,3), - xi(5,1), - xi(4,1),xi(3,3) + xi(1,1) + xi(2,2)))



        [0  0  0  0  0  0]
        [                ]
        [0  0  0  0  0  0]
        [                ]
        [0  0  0  0  0  0]
adx1 := [                ]
        [0  1  0  0  0  0]
        [                ]
        [0  0  1  0  0  0]
        [                ]
        [0  0  0  0  0  0]



        [0   0  0  0  0  0]
        [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [0   0  0  0  0  0]
adx2 := [                 ]
        [-1  0  0  0  0  0]
        [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [0   0  0  0  1  0]



        [0   0  0  0  0  0]
        [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [0   0  0  0  0  0]
adx3 := [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [-1  0  0  0  0  0]
        [                 ]
        [0   0  0  1  0  0]



        [0  0  0   0  0  0]
        [                 ]
        [0  0  0   0  0  0]
        [                 ]
        [0  0  0   0  0  0]
adx4 := [                 ]
        [0  0  0   0  0  0]
        [                 ]
        [0  0  0   0  0  0]
        [                 ]
        [0  0  -1  0  0  0]



        [0  0   0  0  0  0]
        [                 ]
        [0  0   0  0  0  0]
        [                 ]
        [0  0   0  0  0  0]
adx5 := [                 ]
        [0  0   0  0  0  0]
        [                 ]
        [0  0   0  0  0  0]
        [                 ]
        [0  -1  0  0  0  0]


The generic nilpotent derivation : the eigenvalues are 0

xi(1,1):=0

xi(2,2):=0

xi(3,3):=0

by subtracting adjoints one then may suppose:

xi(4,1):=0,xi(4,2):=0,xi(5,1):=0,xi(6,2):=0,xi(6,3):=0

delta:=

[   0        0        0     0  0  0]
[                                  ]
[   0        0        0     0  0  0]
[                                  ]
[   0        0        0     0  0  0]
[                                  ]
[   0        0     xi(4,3)  0  0  0]
[                                  ]
[   0     xi(5,2)     0     0  0  0]
[                                  ]
[xi(6,1)     0        0     0  0  0]


We denote this delta by the shortform

shortformdelta:={xi(4,3),

                 ss,

                 xi(5,2),

                 ss,

                 xi(6,1)}

paramindexeslist:={{4,3},{5,2},{6,1}}

With the first kind automorphism one gets$

shortformdeltaprimemodadg:={(b(2,2)*b(1,1)*xi(4,3))/b(3,3),
ss,
(b(3,3)*b(1,1)*xi(5,2))/b(2,2),
ss,
b(3,3)*b(2,2)*xi(6,1)}$

deltaprimemodg(4,3):=(b(2,2)*b(1,1)*xi(4,3))/b(3,3)$

deltaprimemodg(5,2):=(b(3,3)*b(1,1)*xi(5,2))/b(2,2)$

deltaprimemodg(6,1):=b(3,3)*b(2,2)*xi(6,1)$

det(AUTOM):=b(3,3)**3*b(2,2)**3*b(1,1)**4$

DELTAPRIMEMODADG:=

mat((0,0,0,0,0,0),

    (0,0,0,0,0,0),

    (0,0,0,0,0,0),

          b(2,2)*b(1,1)*xi(4,3)
    (0,0,-----------------------,0,0,0),
                 b(3,3)

        b(3,3)*b(1,1)*xi(5,2)
    (0,-----------------------,0,0,0,0),
               b(2,2)

    (b(3,3)*b(2,2)*xi(6,1),0,0,0,0,0))


With the second kind automorphism one gets$

shortformdeltaprimemodadg:={(bb(2,3)*bb(1,1)*xi(5,2))/bb(3,2),
ss,
(bb(3,2)*bb(1,1)*xi(4,3))/bb(2,3),
ss,
bb(3,2)*bb(2,3)*xi(6,1)}$

deltaprimemodg(4,3):=(bb(2,3)*bb(1,1)*xi(5,2))/bb(3,2)$

deltaprimemodg(5,2):=(bb(3,2)*bb(1,1)*xi(4,3))/bb(2,3)$

deltaprimemodg(6,1):=bb(3,2)*bb(2,3)*xi(6,1)$

det(AUTOM):=bb(3,2)**3*bb(2,3)**3*bb(1,1)**4$

DELTAPRIMEMODADG:=

mat((0,0,0,0,0,0),

    (0,0,0,0,0,0),

    (0,0,0,0,0,0),

          bb(2,3)*bb(1,1)*xi(5,2)
    (0,0,-------------------------,0,0,0),
                  bb(3,2)

        bb(3,2)*bb(1,1)*xi(4,3)
    (0,-------------------------,0,0,0,0),
                bb(2,3)

    (bb(3,2)*bb(2,3)*xi(6,1),0,0,0,0,0))




!With! ! a! suitable! second! kind! autom! ! one! can! permute! xi(4,3)! and! xi
(5,2)! and! le\
ave! xi(6,1)! unchanged.$

*********** SUBCASE 3 :  xi(4,3)=xi(5,2)= 0 *******************$

xi(4,3):=0$

xi(5,2):=0$

Then one necessarily has xi(6,1) neq 0, hence we can suppose xi(6,1):=1$

Hence, we are reduced in the case 3 under consideration to:$

shortformdeltaprime ={0,SS,0,SS,1}$