Diagonalizing $xyz$ The quadratic form  $g(x,y) = xy$    can be diagonalized
by the change of variables  $x = (u + v)$   and  $y = (u - v)$ .
However, it seems unlikely that the cubic form  $f(x,y,z) = xyz$,
can be diagonalized by a linear change of variables.
Is there a short computational or theoretical proof of this?
Thanks.

The "diagonalization" is given on page 101 of this paper:
https://www.sciencedirect.com/science/article/pii/0024379587902898
Here is a picture of the formula:

