The generic automorphism phi of g_{6,15} as computed by calculautom6_15.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),

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

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

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

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

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


                7       7
det(phi):=b(2,2) *b(1,1)

Second kind :

psi:=

mat((0,b(1,2),0,0,0,0),

    (b(2,1),0,0,0,0,0),

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

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

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

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

                                    2       2
     ,b(3,2)*b(2,1)*b(1,2), - b(2,1) *b(1,2) ))


                7       7
det(psi):=b(2,1) *b(1,2)

generic derivation : delta:=

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

    (0,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),0,0),

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

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

     2*(xi(1,1) + xi(2,2))))



        [0  0  0  0  0  0]
        [                ]
        [0  0  0  0  0  0]
        [                ]
        [0  1  0  0  0  0]
adx1 := [                ]
        [0  0  1  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]
        [                 ]
        [-1  0  0  0  0  0]
adx2 := [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [0   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]
adx3 := [                  ]
        [-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  0]
adx4 := [                 ]
        [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  0]
adx5 := [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [-1  0  0  0  0  0]


The generic nilpotent derivation : the eigenvalues are 0

xi(1,1):=0

xi(2,2):=0

by subtracting adjoints one then may suppose:

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

delta:=

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


We denote this delta by the shortform

shortformdelta:={xi(4,2),

                 ss,

                 xi(5,1),

                 xi(5,2)}

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

With the first kind automorphism one gets$

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

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

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

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

det(AUTOM):=b(2,2)**7*b(1,1)**7$

DELTAPRIMEMODADG:=

[       0                   0                      0            0  0  0]
[                                                                      ]
[       0                   0                      0            0  0  0]
[                                                                      ]
[       0                   0                      0            0  0  0]
[                                                                      ]
[                          2                                           ]
[       0            b(1,1) *xi(4,2)               0            0  0  0]
[                                                                      ]
[      2                                                               ]
[b(2,2) *xi(5,1)  b(2,2)*b(1,1)*xi(5,2)            0            0  0  0]
[                                                                      ]
[       0                   0            b(2,2)*b(1,1)*xi(5,2)  0  0  0]


With the second kind automorphism one gets$

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

deltaprimemodg(4,2):= - b(1,2)**2*xi(5,1)$

deltaprimemodg(5,1):= - b(2,1)**2*xi(4,2)$

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

det(AUTOM):=b(2,1)**7*b(1,2)**7$

DELTAPRIMEMODADG:=

[        0                     0                      0            0  0  0]
[                                                                         ]
[        0                     0                      0            0  0  0]
[                                                                         ]
[        0                     0                      0            0  0  0]
[                                                                         ]
[                              2                                          ]
[        0             - b(1,2) *xi(5,1)              0            0  0  0]
[                                                                         ]
[         2                                                               ]
[ - b(2,1) *xi(4,2)  b(2,1)*b(1,2)*xi(5,2)            0            0  0  0]
[                                                                         ]
[        0                     0            b(2,1)*b(1,2)*xi(5,2)  0  0  0]




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

************* CASE 3 : xi(4,2) = xi(5,1) = 0 ******************************$

xi(4,2):=0$

xi(5,1):=0$

Then we necessarily have xi(5,2) NEQ 0, hence we may suppose xi(5,2):=1.$

xi(5,2):=1$

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

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