In Sec. 12 of 'Superstring Theory', by Green, Schwarz, and Witten, the *minimal* spin connection is motivated as follows:

> If we are to avoid modifying the standard content of general
> relativity, the two notions of the covariant derivative of a vector
> $V$ must be equivalent. This will be so, in the sense that
> $D_{\mu}V^{a} = e^{a\nu}D_{\mu}V_{\nu}$, if we define the spin
> connection so that the covariant derivative of the veilbein is zero,
> $D_{\mu}e^{a}_{\nu}=0$.

Until recently, I have always thought that this argument was quite sensible. But now I have doubts, due to the following considerations: The covariant derivative of a vector field $A$, say, in local Lorentz coordinates is given by

$$D_{\mu }A_{a}
\equiv\partial _{\mu }A_{a}-\frac{1}{2}\omega _{\mu cd}\left(
V^{cd}\right) ^{b}{}_{a}A_{b}
=\partial _{\mu }A_{a}-\omega _{\mu }{}^{c}{}_{a}A_{c},$$

where $V^{cd}$ are the generators of the vector representation of the Lorentz group, given by $(V^{cd})^{a}{}_{b} = \eta ^{cb}\delta
_{a}^{d}-\eta ^{db}\delta _{a}^{c}$. But then, using as well the standard expression for $D_{\mu }e^{b}{}_{\nu }$,

\begin{align}
D_{\mu }A_{\nu }
& =
D_{\mu }\left( e^{b}{}_{\nu }A_{b}\right)  \\
&=\left( D_{\mu }e^{b}{}_{\nu }\right) A_{b}+e^{b}{}_{\nu }D_{\mu }A_{b} \\
&=\left( \partial _{\mu }e^{b}{}_{\nu }-\Gamma ^{\rho }{}_{\mu \nu
}e^{b}{}_{\rho }+\omega _{\mu }{}^{b}{}_{c}e^{c}{}_{\nu }\right)
A_{b}+e^{b}{}_{\nu }\left( \partial _{\mu }A_{b}-\omega _{\mu
}{}^{c}{}_{b}A_{c}\right)  \\
&=\left( \partial _{\mu }e^{b}{}_{\nu }\right) A_{b}-\Gamma ^{\rho }{}_{\mu
\nu }e^{b}{}_{\rho }A_{b}+e^{b}{}_{\nu }\left( \partial _{\mu }A_{b}\right) 
\\
&=\partial _{\mu }A_{\nu }-\Gamma ^{\rho }{}_{\mu \nu }A_{\rho } \\
&\equiv \nabla _{\mu }A_{\nu },
\end{align}

where $\Gamma ^{\rho }{}_{\mu \nu}$ are the Christoffel symbols. The derivation depends only on the antisymmetry of the spin connection, not on it being minimal. But if $D_{\mu }A_{\nu }=\nabla _{\mu }A_{\nu }$, as suggested, then the covariant derivative of any tensor field (as tensor products of vector fields) is the standard GR one, and thus the standard content of GR would seem to be unaltered. Am I doing something fundamentally wrong?