Decomposable and Indecomposable Algebras of Degree 8 and Exponent 2

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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(√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 M(t) where the u-invariant of 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.