SO(3,1) has the double cover SL(2,C), which is the symmetry for the pair of Weyl spinors in a Dirac spinor. Dirac gamma matrices contain two copies of SU(2), and this is exactly why the spin group of the Lorentz group is SL(2,C). SO(3,1) is the group of real 4 × 4 matrices, while SL(2,C) is the group of complex 2 × 2 matrices with determinant 1. They arise dynamically, then no gauge group is required to contain them as real forms. That is precisely why SU(4) is viable. Dirac gamma matrices are the SU(2) × SU(2) structure of the Lorentz algebra, because the Dirac’s spinor matrices content are two copies of SU(2). Dirac’s original antimatter prediction was too symmetric; Nature’s stable first generation is chiral and asymmetric; SU(4) vacuum-polarisation dynamics explain this asymmetry and yield the corrected Dirac equation. The writing of this paper is motivated by the kind critical discussion with mathematician Robert A Wilson, who very helpfully pointed out key problems in the mainstream physics approach, as well as critically clarifying the theoretical physics group and lie algebras underlying Dirac’s spinor. (However, any errors in this paper are not due to Dr Wilson, but to the author.)