Tetrad postulate (${\partial }_a{e}^{\mu }{}_b+{\omega }_a{}^{\mu }{}_{\nu }{e}^{\nu }{}_b-{\Gamma }^c{}_{ab}{e}^{\mu }{}_c=0$)

The tetrad postulate expresses compatibility of the spacetime and Lorentz-frame connections on a vierbein. For the inverse coframe $e_{\nu }{}^b$, solving gives ${\omega }_a{}^{\mu }{}_{\nu }=e_{\nu }{}^b({\Gamma }^c{}_{ab}{e}^{\mu }{}_c-{\partial }_a{e}^{\mu }{}_b)$. Metric compatibility makes the lowered connection antisymmetric in its Lorentz indices. Lifting this connection through the Spinor representation of the Lorentz group defines the spinor covariant derivative.