Identities for algebras of matrices over the octonions

This paper describes all the identities of degree 7 satisfied by algebras of 2×2 matrices over the octonions. There are three cases: (1) the full matrix algebra under the usual matrix product, (2) the algebra of Hermitian matrices under the symmetric product, and (3) the algebra of skew-Hermitian matrices under the antisymmetric product. In case (1) we present seven new identities in degree 7 which were discovered by a computer search but which are proved to hold for matrices with entries in any alternative ring. In case (2) we recover the identities of Vasilovsky in degrees 5 and 6 for the special Jordan algebra of a nondegenerate symmetric bilinear form. In case (3) we describe a computational proof that there are no identities in degree 7 which are not implied by anticommutativity.