Identities for Algebras Obtained from the Cayley-dickson Process

The Cayley-Dickson process gives a recursive method of constructing a nonassociative algebra of dimension 2 for all n 0, beginning with any ring of scalars. The algebras in this sequence are known to be flexible quadratic algebras; it follows that they are noncommutative Jordan algebras: they satisfy the flexible identity in degree 3 and the Jordan identity in degree 4. For the integral sedenion algebra (the double of the octonions) we determine a complete set of generators for the multilinear identities in degrees 5. Since these identities are satisfied by all flexible quadratic algebras, it follows that a multilinear identity of degree 5 is satisfied by all the algebras obtained from the Cayley-Dickson process if and only if it is satisfied by the sedenions.