STANDARD ALGEBRAS

In 1948 A. A. Albert defined a standard algebra S by the identities (x, y, z) + (z, x, y) — (x, z, y) = 0 and (x, y, wz) + O, y, xz) + (z, y, wx) = 0. Standard algebras include all associative algebras and commutative Jordan algebras. The radical 9 of any finite-dimensional standard algebra ?! is its maximal nilpotent ideal. It is known that any semisimple standard algebra is a direct sum of simple ideals, and that any simple standard algebra is either associative or a commutative Jordan algebra.In this paper we study Peirce decompositions and derivations of standard algebras.We prove the Wedderburn principal theorem for standard algebras of characteristic 2 (announced in 1950 by A. J. Penico for characteristic 0): if For standard algebras of characteristic0 we prove analogues of the Malcev-Harish-Chandra theorem and the first Whitehead lemma, and we determine when the derivation algebra of S is semisimple. © 1969 by Pacific Journal of Mathematics.