A Theorem on the Structure of Jordan Algebras.

The purpose of this paper is to fill a gap in Albert's structure theory' of abstract Jordan algebras of any characteristic $2, by proving the following: THEOREM A. Let 21 be a finite-dimensional Jordan algebra over a field (. Assume (1) that 21 has an identity element u and (2) that every element of 21 has the form au + z, where a e (D and z is nilpotent. Then 21 = 4u + A3, where Z is a nil subalgebra of 21. This result was proved by Albert for special Jordan algebras and was used by him as a key result in the structure theory of semisimple commutative powerassociative algebras. A question which was left open in Albert's work was that of the structure of simple Jordan algebras over an algebraically closed field having only one nonzero idempotent, the identity u.2 Theorem A, as stated, implies easily that such an algebra is necessarily fu. This result completes the classification of simple Jordan algebras over an arbitrary field. It also permits us to fill several gaps in the representation theory of Jordan algebras which has been developed by the present authors Our proof will be based on two new concepts-inverses and ternary compositionwhich appear to be of some intrinsic interest. Our results on these enable us to adapt Albert's proof of the theorem for special Jordan algebras to the abstract case. An essential step in the proof is a certain ternary identity which we formulated and which has been proved independently by M. Hall and by L. R. Harper, Jr. (their proofs will appear in a forthcoming issue of the Proc. Am. Math. Soc.).