Sedenions, the Clifford algebra Cl(8), and three fermion generations

Recently there has been renewed interest in using tensor products of division algebras, together with their associated Clifford algebras, to identify the structures of the Standard Model. One full generations of leptons and quarks transforming correctly under the electrocolor group SU (3)(c) circle times U(1)(em) can be described in terms of complex octonion algebra C circle times O. By going beyond the division algebras, and considering the larger Cayley-Dickson algebra of sedenions S, this one generation model is extended to exactly three generations. Each generation is contained in an C circle times O subalgebra of C circle times S, however these three subalgebras are not independent of one another. This three generation model can be related to an alternative model of three generations based on the exceptional Jordan algebra J(3)(O). It is speculated that the shared C circle times H algebra common to all three generations might form a basis for CKM mixing.