  In the context of space-time block codes (STBCs), the theory of generalized
quaternion and biquaternion algebras (i.e., tensor products of two quaternion
algebras) over arbitrary base fields is presented, as well as quadratic form
theoretic criteria to check if such algebras are division algebras. For base
fields relevant to STBCs, these criteria are exploited, via Springer's theorem,
to construct several explicit infinite families of (bi-)quaternion division
algebras. These are used to obtain new $2\x 2$ and $4\x 4$ STBCs.
