Krajewski diagrams and spin lifts

A classification of irreducible, dynamically non-degenerate, almost commutative spectral triples is refined. It is extended to include centrally extended spin lifts. Simultaneously it is reduced by imposing three constraints: (i) the condition of vanishing Yang-Mills and mixed gravitational anomalies, (ii) the condition that the fermion representation be complex under the little group, while (iii) massless fermions are to remain neutral under the little group. These constraints single out the standard model with one generation of leptons and quarks and with an arbitrary number of colours.