Noncommutative Jordan algebras of capacity two

Abstract

Let J J be a noncommutative Jordan algebra with 1. If J J has two orthogonal idempotents e e and f f such that 1 = e + f 1 = e + f and such that the Peirce 1 1 -spaces of each are Jordan division rings, then J J is said to have capacity two. We prove that a simple noncommutative Jordan algebra of capacity two is either a Jordan matrix algebra, a quasi-associative algebra, or a type of quadratic algebra whose plus algebra is a Jordan algebra determined by a nondegenerate symmetric bilinear form.