  We prove two results regarding Hodge structures appearing in the cohomology
of complex tori. First, we prove that if a polarizable Hodge structure appears
in the cohomology of a complex torus $T$, it appears in the cohomology of an
abelian variety isomorphic to a subquotient of $T$. Second, we prove a
universality result for the Kuga-Satake construction applied to Hodge
structures of $K3$ type that might not be polarized.
