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.
Decomposable and Indecomposable Algebras of Degree 8 and Exponent 2