The Dirac matrices in curved spacetime are written as $\gamma_{\mu}=e^{a}_{\mu}\gamma_{a}$ where $e^{a}_{\mu}$ are the vielbein fields and $\gamma_{a}$ are the constant Dirac matrices. Given that the Lorentz generators for Dirac spinors can be written as:
$$\sigma_{ab}=\frac{i}{4}[\gamma_{a},\gamma_{b}].$$
would it be correct to say that in an infinitesimally flat local frame of a curved spacetime one can define the Dirac spinor Lorentz generators as follows?
$$\sigma_{\mu\nu}=\frac{i}{4}e^{a}_{\mu}e^{b}_{\nu}[\gamma_{a},\gamma_{b}].$$
If the above is not correct, how can one define the Lorentz generators in an infinitesimally flat local frame of a curved spacetime?
