The generic automorphism phi of g_{6,20} as computed by calculautom6_20.red :

phi:=

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

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

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

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

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

    (b(6,1),b(6,2),

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

                                             2                  4       7
     b(1,1)*( - b(3,2)*b(3,1) + b(4,1)*b(1,1) ), - b(3,1)*b(1,1) ,b(1,1) ))


                22
det(phi):=b(1,1)

generic derivation : delta:=

[xi(1,1)      0          0           0          0           0    ]
[                                                                ]
[   0     2*xi(1,1)      0           0          0           0    ]
[                                                                ]
[xi(3,1)   xi(3,2)   3*xi(1,1)       0          0           0    ]
[                                                                ]
[xi(4,1)      0       xi(3,2)    4*xi(1,1)      0           0    ]
[                                                                ]
[xi(5,1)   xi(5,2)    - xi(3,1)   xi(3,2)   5*xi(1,1)       0    ]
[                                                                ]
[xi(6,1)   xi(6,2)    - xi(5,1)   xi(4,1)    - xi(3,1)  7*xi(1,1)]



        [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  1  0  0]
        [                ]
        [0  0  0  0  0  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  0  1  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  -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]
        [                 ]
        [-1  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

by subtracting adjoints one then may suppose:

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

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

With the generic automorphism one gets$

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

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

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

det(AUTOM):=b(1,1)**22$

DELTAPRIMEMODADG:=

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


****************** CASE 1 : xi(5,2) NEQ 0 *************************$

Then one may suppose xi(5,2):=1 and one gets deltaprimemodg(4,2)=k by taking :$

xi(5,2):=1$

b(1,1):=k**(1/3)$

With the generic automorphism one gets$

shortformdeltaprimemodadg:={k,ss,xi(6,1)*k**2}$

deltaprimemodg(5,2):=k$

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

det(AUTOM):=k**(1/3)*k**7$

DELTAPRIMEMODADG:=

[    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       k  0  0  0  0]
[                         ]
[         2               ]
[xi(6,1)*k   0  0  0  0  0]


If xi(6,1) NEQ 0, we get deltaprime(6,1):=1 by taking :$

k:=1/xi(6,1)$

With the generic automorphism one gets$

shortformdeltaprimemodadg:={1/xi(6,1),ss,1/xi(6,1)}$

deltaprimemodg(5,2):=1/xi(6,1)$

deltaprimemodg(6,1):=1/xi(6,1)$

det(AUTOM):=(1/xi(6,1))**(1/3)/xi(6,1)**7$

DELTAPRIMEMODADG:=

[    0          0      0  0  0  0]
[                                ]
[    0          0      0  0  0  0]
[                                ]
[    0          0      0  0  0  0]
[                                ]
[    0          0      0  0  0  0]
[                                ]
[               1                ]
[    0      ---------  0  0  0  0]
[            xi(6,1)             ]
[                                ]
[    1                           ]
[---------      0      0  0  0  0]
[ xi(6,1)                        ]


Hence, we are reduced in the case 1 under consideration to:$

shortformdeltaprime ={1,SS,epsilon}$

where epsilon=xi(6,1) =0,1.$

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

In that case, as we suppose delta NEQ 0 one necessarily has xi(6,1) NEQ 0$

Hence we can suppose xi(6,1):=1,$

and we are reduced  to:$

shortformdeltaprime ={0,SS,1}.$