, the Lie algebra of all endomorphisms
of the vector space 
.
We shall use its basis 
for 
,
where 
stands for the matrix with 1 at entry i,j and
0 elsewhere.
For simplicity of computations, we choose a basis
x,y,z,t of 
and let 
act
on 
by the usual matrix action on column vectors
from the left.
Thus, for instance
This then determines the action of 
on 
and hence the action on V.
For example,
As a basis for V, we shall use the 16 monomials in x,y,z,t
which are not cubes.
Let us denote by 
the 
-th monomial in the second row of the
array
below.
The action of 
on V can now be explicitly given
in terms of the following matrix 
where 
is written as
Here 
is the 
matrix
whose ij-th row consists of the coefficients of
written out on the basis of V just given.
In the array below, the row beginning with 
represents
the
ij-th row of 
, that is, the
vector 
such that
.
For 
, we write 
to denote the Lie subalgebra of 
consisting of all 
with 
.
A major tool for discriminating orbits is the following
family of varieties 
:
If f and 
are equivalent by an element y in G,
then 
and so the dimensions of
and 
are the same. Hence all forms equivalent
to a given f are in the same 
. Suppose 
is the
stabilizer in G of a form f and 
is its Lie algebra
in 
. Then 
is a Lie subalgebra of 
. This
gives that the
dimension of 
is at most the dimension of 
. Therefore,
the dimension of the orbit Gf of f is at least i if
f is in 
.
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