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This is a study of algebras with involution that become isomorphic over aseparable closure of the base field to a tensor product of two compositionalgebras. We classify these algebras, provide criteria for isomorphism andisotopy, and determine their automorphism groups, structure groups, andnorm-similitude groups. The most interesting cases of these algebras are the$64$-dimensional bi-octonion algebras, on which groups of absolute type $(G_2\times G_2)\rtimes \mathbb{Z}/2\mathbb{Z}$ and $\mathbf{Spin}_{14}$ actfaithfully by automorphisms and norm-preserving isotopies, respectively. Weclassify the cohomological invariants of these algebras, characterise when theyare division algebras, and study their 14-dimensional Albert forms and64-dimensional octic norms. The main application that we reach is aclassification of the mod 2 cohomological invariants of 14-dimensionalquadratic forms in $I^3$, and of the group $\mathbf{Spin}_{14}$. This extendsGaribaldi's classification of cohomological invariants for lower-dimensionalSpin groups.

Galois cohomology of algebraic groups acting on the tensor product of two composition algebras