  It shown that if a vector space carries commuting actions of two Clifford
algebras, then the quadratic monomials using generators from either Clifford
algebra determine a spinor representation of an orthogonal Lie algebra.
  Examples of this construction have applications to high energy physics,
particularly to the standard model and unification. It is shown how to use
Clifford data to construct spectral triples for the Pati-Salam model that admit
an action of Spin(10).
