Decomposable and Indecomposable Algebras of Degree 8 and Exponent 2

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.