A classification theorem for Albert algebras

Let $k$ be a field whose characteristic is different from 2 and 3 and let $L/k$ be a quadratic extension. In this paper we prove that for a fixed, degree 3 central simple algebra $B$ over $L$ with an involution $\sigma$ of the second kind over $k$, the Jordan algebra $J(B,\sigma,u,\mu)$, obtained through Tits' second construction is determined up to isomorphism by the class of $(u,\mu)$ in $H^1(k,SU(B,\sigma))$, thus settling a question raised by Petersson and Racine. As a consequence, we derive a ``Skolem Noether'' type theorem for Albert algebras. We also show that the cohomological invariants determine the isomorphism class of $J(B,\sigma,u,\mu)$, if $(B,\sigma)$ is fixed.