  The Kuga-Satake construction associates to a K3 type polarized weight 2 Hodge
structure H an abelian variety A such that H is a quotient Hodge structure of
H^2(A). The first step is to consider the Clifford algebra of H. It turns out
that it is endowed with a weight 2 Hodge structure compatible with the algebra
structure. We show more generally that a weight 2 polarized Hodge structure
which carries a compatible (unitary, associative) algebra structure is a
quotient of the H^2 of an abelian variety.
