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 $V \subsetneq K^n$
and require a Hurwitz-type ``bilinear condition'' on $V$. This
means the existence of a bilinear map
$\varphi\colon K^n \times K^n \rightarrow K^n$
such that $\varphi(V \times V) \subset V$
and $q(\mathbf{X}) q(\mathbf{Y}) = q(\varphi(\mathbf{X}, \mathbf{Y}))$
for any $\mathbf{X}, \mathbf{Y} \in V$. We show
that the $m$-fold Pfister form is multiplicative on certain
proper subvariety in $K^{2^m}$ 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.