  We study the decomposition of central simple algebras of exponent 2 into
tensor products of quaternion algebras. We consider in particular
decompositions in which one of the quaternion algebras contains a given
quadratic extension. Let $B$ be a biquaternion algebra over $F(\sqrt{a})$ with
trivial corestriction. A degree 3 cohomological invariant is defined and we
show that it determines whether $B$ has a descent to $F$. This invariant is
used to give examples of indecomposable algebras of degree 8 and exponent 2
over a field of 2-cohomological dimension 3 and over a field $\mathbb M(t)$
where the $u$-invariant of $\mathbb M$ is 8 and $t$ is an indeterminate. The
construction of these indecomposable algebras uses Chow group computations
provided by A. S. Merkurjev in Appendix.
