A VARIETY CONTAINING JORDAN AND PSEUDO-COMPOSITION ALGEBRAS

Abstract

We consider 3-Jordan algebras, i.e., the nonassociative commutative algebras satisfying (x3y)x = x3(yx). The variety of 3-Jordan algebras contains all Jordan algebras and all pseudo-composition algebras. We prove that a simple 3-Jordan algebra with idempotent is either a Jordan algebra or a pseudo-composition algebra.