NONCOMMUTATIVE MATRIX JORDAN ALGEBRAS, CAYLEY-DICKSON ALGEBRAS, AND SCHAFERS THEOREM

We provide a construction of noncommutative Jordan algebras of degree two. The construction can be iterated, and pre show that after the first few iterations no new derivations arise. The relationship between this iterative process and the Cayley-Dickson process is studied, and the result on derivations is used to obtain a generalization of Schafer's classical theorem on the derivation algebras of Cayley-Dickson algebras.