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In this section we show that any form in three variables modulo the cubics is equivalent to

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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