I asked this question https://physics.stackexchange.com/questions/820924/is-tetrad-postualte-independent-of-gauge-field

Here is what I know, $g_{\mu \nu} = e^{a}_{\mu} e^{b}_{\nu} \eta_{ab}$ and the tetrad postulate is \begin{equation} D_{\rho} e^{a}_{\nu} = \partial_{\rho} e^{a}_{\mu} - \Gamma^{\lambda}_{\rho \nu}e^{a}_{\lambda}+ \omega^{a}_{b \mu} e^{b}_{\nu}=0 \end{equation} But in case of abelian or non abelian Yang mills, $D_{\mu}$ is different and that should affect the tetrad postulate. In the same spirit what becomes of the statement that gamma matrices are covariantly constant. The covariant derivative in flat space-time for abelian gauge field is $D = \gamma^{\mu}(\partial_{\mu} + iA_{\mu})$ and $D \gamma^{\nu} =0$ here using $\partial_{\mu}\gamma^{\nu}$ we get $\gamma^{\mu} A_{\mu} \gamma^{\nu}=0$ what does it mean? I don't understand the underlying mathematical structure.

The answer I received is that objects inavriant under gauge transformation, their covariant derivative does not contain gauge field. It makes sense but now I am looking for a more precise explanation about the gauge bundle we are working with and how connections act on the relevant objects.