Internal Symmetry from Division Algebras in Pure Spinor Geometry

The É. Cartan’s equations defining “simple” spinors (renamed “pure” by C. Chevalley) are interpreted as equations of motions for fermion multiplets in momentum spaces which, in a constructive approach based bilinearly on those spinors, result compact and Lorentzian, naturally ending up with a ten-dimension space. The equations found are most of those traditionally adopted ad hoc by theoretical physics in order to represent the observed phenomenology of elementary particles. In particular, it is shown how the known internal symmetry groups, in particular those of the standard model, might derive from the 3 complex division algebras correlated with the associated Clifford algebras. They also explain the origin of charges, the tendency of fermions to appear in charged-neutral doublets, as well as the origin of families. The adoption of the Cartan’s conjecture on the non-elementary nature of Euclidean geometry (bilinearly generated by simple or pure spinors) might throw light on several problematic aspects of particle physics.