On a homomorphism property of certain Jordan algebras

Abstract

This result has a number of important consequences. It may be seen to imply that no simple exceptional finite dimensional Jordan algebra of characteristic not two is a homomorphic image of a special(2) Jordan algebra. But there is an exceptional simple Jordan algebra S& (of dimension 27) over a field a of characteristic not two which is generated by three of its elements. Then ID is a homomorphic image of the free Jordan algebra '3 on three generators. It follows that '3 is not special, and we also know that '3 is not isomorphic to the free special Jordan algebra 3 [x, y, z, 1 ] consisting of all reversible polynomials(') in the free associative algebra a [x, y, , 1Z] of all polynomials on the three generators x, y, z. 2. Elementary properties of 60 and certain subalgebras. We begin with a brief description of the algebra eo of our theorem. Let Z be an algebra with a unity element e over a field 0 of characteristic not two. Suppose that Z has an involution T over a; that is, an antiautomorphism