Octonion algebras obtained from associative algebras with involution (Q2781320)

In 1978, \textit{S. Okubo} [Hadronic J. 1, 1250-1278 (1978; Zbl 0417.17011)] showed that the space of \(3\times 3\) traceless complex matrices can be endowed with a multiplication, derived from the usual matrix multiplication, in such a way that it becomes a nonunital composition algebra. This construction was rediscovered in a more general setting by \textit{J. R. Faulkner} [Proc. Am. Math. Soc. 104, 1027-1030 (1988; Zbl 0698.17004)], and studied since then by a number of authors. The composition algebras thus obtained are examples of symmetric composition algebras (see Chapter 8 of the book by \textit{M.-A. Knus, A. Merkurjev, M. Rost} and \textit{J.-P. Tignol} [The Book of Involutions, AMS, Providence, RI (1998; Zbl 0955.16001)]. NEWLINENEWLINENEWLINESince any eight-dimensional composition algebra determines, up to isomorphism, a unique octonion algebra, the Okubo-Faulkner construction can be interpreted as a way of obtaining octonion algebras from central simple degree \(3\) associative algebras. On the other hand, there is another construction due to \textit{D. E. Haile, M.-A. Knus, M. Rost} and \textit{J.-P. Tignol} [Isr. J. Math. 96, Part B, 299-340 (1996; Zbl 0871.16017)], attaching a \(3\)-fold Pfister form (the norm of an octonion algebra) to any central simple associative algebra \((B,*)\) of degree \(3\) with involution of the second kind. NEWLINENEWLINENEWLINEIn the paper under review, a different and very nice approach is used to explicitly obtain an octonion algebra from any such \((B,*)\) over fields of characteristic \(\neq 2,3\). An approach that uses exceptional central simple Jordan algebras, or Albert algebras. The idea behind the construction is to pass, by means of the Tits process, from \((B,*)\) to an Albert algebra \(J\), which is reduced and hence isomorphic to an algebra of Hermitian \(3\times 3\) matrices over an octonion algebra, and then to determine this octonion algebra from the original data. NEWLINENEWLINENEWLINEThe new approach is related to the previous ones mentioned above.