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We study the decomposition of central simple algebras of exponent 2 intotensor products of quaternion algebras. We consider in particulardecompositions in which one of the quaternion algebras contains a givenquadratic extension. Let $B$ be a biquaternion algebra over $F(\sqrt{a})$ withtrivial corestriction. A degree 3 cohomological invariant is defined and weshow that it determines whether $B$ has a descent to $F$. This invariant isused to give examples of indecomposable algebras of degree 8 and exponent 2over 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. Theconstruction of these indecomposable algebras uses Chow group computationsprovided by A. S. Merkurjev in Appendix.

Decomposable and Indecomposable Algebras of Degree 8 and Exponent 2