In this paper we will classify the finite spectral triples with KO-dimension six, following the classification found in [1,2,3,4], with up to four summands in the matrix algebra. Again, heavy use is made of Krajewski diagrams . Furthermore we will show that any real finite spectral triple in KO-dimension 6 is automatically S 0 -real. This work has been inspired by the recent paper by Alain Connes  and John Barrett .
In the classification we find that the standard model of particle physics in its minimal version fits the axioms of noncommutative geometry in the case of KO-dimension six. By minimal version it is meant that at least one neutrino has to be massless and mass-terms mixing particles and antiparticles are prohibited