Abstract Metric-affine gravity (GL (4) gauge theory) in 4-dimensions is coupled to a spacetime Dirac source field using the isomorphisms of the Lie algebra gl (4) to the Clifford algebras Cl (3, 1) and Cl (2, 2). A simple transformation relates the generators of Cl (3, 1) to a real representation of Cl (2, 2), while the real representation of Cl (2, 2) serves directly as a basis for the Lie algebra gl (4). Therefore, although GL (4) does not contain a spinor representation of the Lorentz group, expanding its Lie algebra in the Cl (2, 2) basis gives a Clifford valued connection with well-defined coupling to Dirac spinors. Variation of the expansion coefficients gives new Dirac sources for both torsion and nonmetricity, separated by identifying the so (3, 1) 3, 1 basis within the gl (4) basis.
James T. Wheeler (Wed,) studied this question.