Generalized Clifford algebras and hyperspin manifolds

We consider a special extension of Clifford algebras and show that these generalized Clifford algebras are naturally equipped with a metric defined by a fundamental form of degreen which isSL(n, φ) ⊗SL(n,φ) invariant. Using the embedding of the quaternions in the generalized Clifford algebras, in the Hermitian limit, we obtain an algebraic description of the inclusion of the Minkowski space into the hyperspin manifold.