  The tensor product of the division algebras, which is a kernel for the
structure of the Standard Model, is also a root for the Clifford algebra of
(1,9)-space-time. A conventional Dirac Lagrangian, employing the (1,9)-Dirac
operator acting on the Standard Model hyperfield, gives rise to matter into
antimatter transitions not mediated by any gauge field. These transitions are
eliminated by restricting the dependencies of the components of the hyperfield
on the extra six dimensions, which appear in this context as a complex triple.
