A note on reduced Jordan algebras

Abstract

In this paper we give short proofs of two of the main theorems concerning reduced exceptional Jordan algebras A = H(C3, 'y): the AlbertJacobson Theorem that the Cayley coordinate algebra C is determined by A up to isomorphism, and the Springer Theorem that two such algebras A, A' are isomorphic if and only if they have isomorphic coordinate algebras and equivalent trace forms. We avoid using the generic norm by working directly with the reduced idempotents. Our proofs do not require that the algebras be exceptional, and are valid for arbitrary reduced simple algebras. 1. Reduced idempotents. Throughout this paper, A will denote a Jordan algebra with identity element 1 over a field 1 of characteristic 5 2. In this section we make no assumptions about the simplicity or finite-dimensionality of A. Recall that an idempotent e is reduced if Ai(e) = UeA =cJe. We assume the reader is familiar with the operators U, = 2L' -Lx2and the basic identities involving them (e.g. [3,