A Cayley-Dickson Process for a Class of Structurable Algebras

Abstract

In this paper, we study the class of all simple structurable algebras with the property that the space of skew-hermitian elements has dimension $1$. These algebras with involution have arisen in the study of Lie algebra constructions. The reduced algebras are isotopic to $2 \times 2$ matrix algebras. We study a Cayley-Dickson process for rationally constructing some algebras in the class including division algebras and nonreduced nondivision algebras. An important special case of the process endows the direct sum of two copies of a $28$-dimensional degree $4$ central simple Jordan algebra $\mathcal {B}$ with the structure of an algebra with involution. In preparatory work, we obtain a procedure for giving the space ${\mathcal {B}_0}$ of trace zero elements of any such Jordan algebra $\mathcal {B}$ the structure of a $27$-dimensional exceptional Jordan algebra.