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.