= Spinor covariant derivative
{title2=$\nabla_\mu\psi=\partial_\mu\psi+\tfrac14\omega_{\mu AB}\gamma^{AB}\psi$}
Lift the <Levi-Civita connection> to the <spinor bundle> using the spin representation of the Lorentz algebra. Under a <local Lorentz transformation>, its matrix $\Omega_\mu=\tfrac14\omega_{\mu AB}\gamma^{AB}$ obeys $\Omega'_\mu=S\Omega_\mu S^{-1}-(\partial_\mu S)S^{-1}$, so $\nabla'_\mu\psi'=S\nabla_\mu\psi$. 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 $\partial_\mu\gamma^\nu+\Gamma^\nu_{\mu\rho}\gamma^\rho+[\Omega_\mu,\gamma^\nu]=0$, establishing compatibility with curved Clifford multiplication.
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