, or 
. Using the notation
introduced earlier we may assume that f is
If all coefficients are 0 except 
, we are done. This
means one of the other terms is not 0 and we may assume
and by scaling assume 
. Now set
to get an equivalent form with 
. We now
divide into two cases depending on whether or not 
.
Case 1: 
.
Now set 
.
Let 
and 
be a root of 
. The
resulting form has 
.
If 
, the form is
. If 
, this is 
which is on the
list. Otherwise scale to get 
. This is
which by interchanging variables is 
which
is on the list.
If 
, scale to get 
. The resulting form is
. Transforming xyz by
gives this form.
Case 2: 
.
Scale to get 
. Now transform this by
. Set 
and 
a root
of 
to get 
. The form is now
. If 
,
this is Case 1 with the variables permuted according to
. If not it can be scaled to give
. If 
, this is 
. Now
transform 
by 
and then by
to get it.
Assume then that 
. The form is now
. Let a,b,c be the three distinct
roots of 
where 
.
Now transform the form xyz by
to get the form
. By choosing
appropriately these are then all equivalent.
This transformation is found as
in by finding a linear transformation which under
conjugation transforms the Lie algebras for the centralizers of the
forms. The coeffients for the transformed xyz are obtained using
the facts that abc=1, a+b+c=0,
and also that
. All but the last follow
immediately because a,b,c are the roots of the equation
. The last follows because 
and 
are the two roots of 
. The sum of
these terms, 0, can be found
from computing the coefficient
of 
in 
. The product, 
, can be evaluated
using the expressions above.
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