Twisted Reality and the Second-Order Condition

An interesting feature of the finite-dimensional real spectral triple (A,H,D,J) Standard Model is that it satisfies a ``second-order'' condition: conjugation by J maps Clifford algebra Cl_D(A) into its commutant, which in fact isomorphic to itself (H self-Morita equivalence Cl_D(A)-bimodule). This resembles property canonical closed oriented Riemannian manifold: there dense subspace H Cl_D(A)-bimodule. In this paper we argue on manifolds, order for be implemented reality operator J, one has introduce ``twist'' and weaken axioms triples. We then investigate how above mentioned conditions behave under products