General Representation Theory of Jordan Algebras

The theory of Jordan algebras has originated in the study of subspaces of an associative algebra that are closed relative to the composition ab=a×b+b×a where the × denotes the associative product. Such systems are called special Jordan algebras. It is well known that the composition ab satisfies the conditions 0.1 $$ ab = ba,\;\left( {{a^2}b} \right)a = {a^2}\left( {ba} \right) $$ This has led to the definition of an (abstract) Jordan algebra as a (nonassociative) algebra whose multiplication satisfies the above conditions. It is an open question as to how extensive is the subclass of special Jordan algebras in the class of Jordan algebras. However, it is known that there exist Jordan algebras which are not special.