Gravitation Page 276 Exercise 11.3 solution indicated that

$$\nabla_\gamma \nabla _\delta e_\beta =e_{\mu}\Gamma^\mu_{\beta\delta,\gamma} +(e_\nu\Gamma^\nu_{\mu\gamma}) \Gamma^\mu_{\beta\delta}$$

However, using to "Chain rule for gradient" i.e. Eq 10.22 $$\nabla_\gamma \nabla _\delta e_\beta =\nabla_{\gamma}(e_{\mu}\Gamma^\mu_{\beta\delta,\gamma}) =e_{\mu} \Gamma^\mu_{\beta\delta;\gamma} +(e_{\nu;\gamma}) \Gamma^\mu_{\beta\delta}$$

where according to equation 10.19, $$\Gamma^\mu_{\beta\delta;\gamma} $$ did have indices and have to be corrected.

Yet the answer somehow indicated that $$\Gamma^\mu_{\beta\delta;\gamma}=\Gamma^\mu_{\beta\delta,\gamma} $$.

Why $\Gamma^\mu_{\beta\delta;\gamma} =\Gamma^\mu_{\beta\delta,\gamma}$? Was there some identity? How to show that $\Gamma^\mu_{\beta\delta;\gamma} =\Gamma^\mu_{\beta\delta,\gamma}$?

Gravitation Page 276 Exercise 11.3 solution indicated that

$$\nabla_\gamma \nabla _\delta e_\beta =e_{\mu}\Gamma^\mu_{\beta\delta,\gamma} +(e_\nu\Gamma^\nu_{\mu\gamma}) \Gamma^\mu_{\beta\delta}$$

However, using to "Chain rule for gradient" i.e. Eq 10.22 $$\nabla_\gamma \nabla _\delta e_\beta =\nabla_{\gamma}(e_{\mu}\Gamma^\mu_{\beta\delta,\gamma}) =e_{\mu} \Gamma^\mu_{\beta\delta;\gamma} +(e_{\mu;\gamma}) \Gamma^\mu_{\beta\delta}$$

where according to equation 10.19, $$\Gamma^\mu_{\beta\delta;\gamma} $$ did have indices and have to be corrected.

Yet the answer somehow indicated that $$\Gamma^\mu_{\beta\delta;\gamma}=\Gamma^\mu_{\beta\delta,\gamma} $$.

Why $\Gamma^\mu_{\beta\delta;\gamma} =\Gamma^\mu_{\beta\delta,\gamma}$? Was there some identity? How to show that $\Gamma^\mu_{\beta\delta;\gamma} =\Gamma^\mu_{\beta\delta,\gamma}$?