 Spinor covariant derivative

Lift the Levi-Civita connection to the spinor bundle using the spin representation of the Lorentz algebra. Under a local Lorentz transformation, its matrix ${\Omega }_{\mu }=\frac{1}{4}{\omega }_{\mu AB}{\gamma }^{AB}$ obeys ${\Omega }_{\mu }^{\prime }=S{\Omega }_{\mu }{S}^{-1}-({\partial }_{\mu }S)S^{-1}$, so ${\nabla }_{\mu }^{\prime }{\psi }^{\prime }=S{\nabla }_{\mu }\psi$. The Clifford algebra identity $[{\gamma }^C,{\gamma }^{AB}]=2({\eta }^{CA}{\gamma }^B-{\eta }^{CB}{\gamma }^A)$ fixes the coefficient. The tetrad postulate then gives ${\partial }_{\mu }{\gamma }^{\nu }+{\Gamma }_{\mu \rho }^{\nu }{\gamma }^{\rho }+[{\Omega }_{\mu },{\gamma }^{\nu }]=0$, establishing compatibility with curved Clifford multiplication.