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=

Set , 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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