Multiplicative quadratic forms on algebraic varieties

In this note we extend Hurwitz-type multiplication of quadratic forms. For a regular quadratic space (K-n, q), we restrict the domain of q to an algebraic variety V subset of or equal to K-n and require a Hurwitz-type "bilinear condition" on V. This means the existence of a bilinear map phi: K-n x K-n --> K-n such that phi(V x V) subset of V and q(X)q(Y) = q(phi(X, Y)) for any X, Y is an element of V. We show that the m-fold Pfister form is multiplicative on certain proper subvariety in K-2m for any m. We also show the existence of multiplicative quadratic forms which are different from Pfister forms on certain algebraic varieties for n = 4, 6. Especially for n = 4 we give a certain family of them.