The Wedderburn principal theorem for Jordan algebras

If a subspace W of ?* is closed with respect to this new multiplication, then W is a Jordan algebra relative to the new multiplication. A special Jordan algebra(2) is a Jordan algebra isomorphic to one obtained from an associative algebra in the above manner. The title of this paper is the same as that of a paper by A. A. Albert [3]. However, in that paper Albert considers only special Jordan algebras, while we prove the principal theorem here for the general Jordan algebras defined by commutativity and the identity (1.1). In each case the base field is assumed to be of characteristic 0. Considering the Jordan algebra Q3, we define(3)