(x12 +x22 )(y12 + y22) = (x1y1 - x2y2)2+ (x1y2 + x2y1)2
This formula might be generalized as
(x12 + ... + xr2)(y12 + ... + ys2) = z12 + ... + zn2
where each zi is "bilinear" in the x's and y's in the sense that it is a sum of monomials of the form c (xi yk). These identities are relevant to questions about normed algebras, embeddings of topological spaces, and linear algebra.
We'll find a few examples of such identities, but the problem of finding this type of identity is extremely difficult. It turns out to be easier to show that identities cannot exist under certain circumstances.