Short question that could potentially be answered without reading all the details:
Is ${\Gamma^\mu}_{\sigma\rho}{A^\sigma}_{,\nu}={\Gamma^\mu}_{\sigma\nu}{A^\sigma}_{,\rho}$? If so, why? If not, what has gone wrong with the following derivation?
Motivation for this question: I'm reading a text that derives the components of the Riemann tensor by examining the second covariant derivative of a vector. The first covariant derivative is defined as
\begin{equation} D_\nu A^\mu\equiv\partial_\nu A^\mu+{\Gamma^\mu}_{\rho\nu}A^\rho \end{equation}
and the second covariant derivative is worked out to be
\begin{equation} D_\rho(D_\nu A^\mu)=({A^\mu}_{,\nu}+{\Gamma^\mu}_{\sigma\nu}A^\sigma)_{,\rho}+{\Gamma^\mu}_{\sigma\rho}({A^\sigma}_{,\nu}+{\Gamma^\sigma}_{\tau\nu}A^\tau)-{\Gamma^\sigma}_{\nu\rho}({A^\mu}_{,\sigma}+{\Gamma^\mu}_{\tau\sigma}A^\tau) \end{equation}
The author then skips a few steps and arrives that the result that
\begin{equation} (D_\rho D_\nu -D_\nu D_\rho)A^\mu={R^\mu}_{\nu\tau\rho}A^\tau \end{equation}
where ${R^\mu}_{\nu\tau\rho}$ are the components of the Riemann tensor:
\begin{equation} {R^\mu}_{\nu\tau\rho}\equiv \partial_\rho {\Gamma^\mu}_{\nu\tau} - \partial_\nu {\Gamma^\mu}_{\rho\tau} +{\Gamma^\mu}_{\rho\sigma}{\Gamma^\sigma}_{\nu\tau}-{\Gamma^\mu}_{\nu\sigma}{\Gamma^\sigma}_{\rho\tau} \end{equation}
I am trying to fill in the steps to verify the expression for ${R^\mu}_{\nu\tau\rho}$. This is what I have so far:
\begin{align} (D_\rho D_\nu -D_\nu D_\rho)A^\mu &=[({A^\mu}_{,\nu}+{\Gamma^\mu}_{\sigma\nu}A^\sigma)_{,\rho}+{\Gamma^\mu}_{\sigma\rho}({A^\sigma}_{,\nu}+{\Gamma^\sigma}_{\tau\nu}A^\tau)-{\Gamma^\sigma}_{\nu\rho}({A^\mu}_{,\sigma}+{\Gamma^\mu}_{\tau\sigma}A^\tau)]- [({A^\mu}_{,\rho}+{\Gamma^\mu}_{\sigma\rho}A^\sigma)_{,\nu}+{\Gamma^\mu}_{\sigma\nu}({A^\sigma}_{,\rho}+{\Gamma^\sigma}_{\tau\rho}A^\tau)-{\Gamma^\sigma}_{\rho\nu}({A^\mu}_{,\sigma}+{\Gamma^\mu}_{\tau\sigma}A^\tau)]\\ &=[\color{blue}{{A^\mu}_{,\nu,\rho}}+\partial_\rho({\Gamma^\mu}_{\sigma\nu}A^\sigma)+{\Gamma^\mu}_{\sigma\rho}{A^\sigma}_{,\nu}+{\Gamma^\mu}_{\sigma\rho}{\Gamma^\sigma}_{\tau\nu}A^\tau- \color{red}{{\Gamma^\sigma}_{\nu\rho}{A^\mu}_{,\sigma}}-\color{green}{{\Gamma^\sigma}_{\nu\rho}{\Gamma^\mu}_{\tau\sigma}A^\tau}] -[\color{blue}{{A^\mu}_{,\rho,\nu}}+\partial_\nu({\Gamma^\mu}_{\sigma\rho}A^\sigma)+{\Gamma^\mu}_{\sigma\nu}{A^\sigma}_{,\rho}+{\Gamma^\mu}_{\sigma\nu}{\Gamma^\sigma}_{\tau\rho}A^\tau- \color{red}{{\Gamma^\sigma}_{\rho\nu}{A^\mu}_{,\sigma}}-\color{green}{{\Gamma^\sigma}_{\rho\nu}{\Gamma^\mu}_{\tau\sigma}A^\tau}] \end{align}
Now, some things cancel because we're allowed to exchange the order on second partial derivatives. I've colored the pairs that I think cancel this way.
Everything here matches the expression for ${R^\mu}_{\nu\tau\rho}$ except for the two terms ${\Gamma^\mu}_{\sigma\rho}{A^\sigma}_{,\nu}$ and ${\Gamma^\mu}_{\sigma\nu}{A^\sigma}_{,\rho}$. If these terms are equal, then they cancel and we are done. Are they equal? If so, why? If not, can you see what I have done wrong?
As Anthony pointed out, I did not yet apply the product rule to some of my terms:
\begin{align} (D_\rho D_\nu -D_\nu D_\rho)A^\mu &=[\color{blue}{{A^\mu}_{,\nu,\rho}}+\partial_\rho({\Gamma^\mu}_{\sigma\nu}A^\sigma)+{\Gamma^\mu}_{\sigma\rho}{A^\sigma}_{,\nu}+{\Gamma^\mu}_{\sigma\rho}{\Gamma^\sigma}_{\tau\nu}A^\tau- \color{red}{{\Gamma^\sigma}_{\nu\rho}{A^\mu}_{,\sigma}}-\color{green}{{\Gamma^\sigma}_{\nu\rho}{\Gamma^\mu}_{\tau\sigma}A^\tau}] -[\color{blue}{{A^\mu}_{,\rho,\nu}}+\partial_\nu({\Gamma^\mu}_{\sigma\rho}A^\sigma)+{\Gamma^\mu}_{\sigma\nu}{A^\sigma}_{,\rho}+{\Gamma^\mu}_{\sigma\nu}{\Gamma^\sigma}_{\tau\rho}A^\tau- \color{red}{{\Gamma^\sigma}_{\rho\nu}{A^\mu}_{,\sigma}}-\color{green}{{\Gamma^\sigma}_{\rho\nu}{\Gamma^\mu}_{\tau\sigma}A^\tau}]\\ &=[\partial_\rho({\Gamma^\mu}_{\sigma\nu}A^\sigma)+{\Gamma^\mu}_{\sigma\rho}{A^\sigma}_{,\nu}+{\Gamma^\mu}_{\sigma\rho}{\Gamma^\sigma}_{\tau\nu}A^\tau] -[\partial_\nu({\Gamma^\mu}_{\sigma\rho}A^\sigma)+{\Gamma^\mu}_{\sigma\nu}{A^\sigma}_{,\rho}+{\Gamma^\mu}_{\sigma\nu}{\Gamma^\sigma}_{\tau\rho}A^\tau]\\ &=[{\Gamma^\mu}_{\sigma\nu,\rho}A^\sigma+\color{orange}{{\Gamma^\mu}_{\sigma\nu}{A^\sigma}_{,\rho}}+\color{purple}{{\Gamma^\mu}_{\sigma\rho}{A^\sigma}_{,\nu}}+{\Gamma^\mu}_{\sigma\rho}{\Gamma^\sigma}_{\tau\nu}A^\tau] -[{\Gamma^\mu}_{\sigma\rho,\nu}A^\sigma+\color{purple}{{\Gamma^\mu}_{\sigma\rho}{A^\sigma}_{,\nu}}+\color{orange}{{\Gamma^\mu}_{\sigma\nu}{A^\sigma}_{,\rho}}+{\Gamma^\mu}_{\sigma\nu}{\Gamma^\sigma}_{\tau\rho}A^\tau]\\ &={\Gamma^\mu}_{\sigma\nu,\rho}A^\sigma+{\Gamma^\mu}_{\sigma\rho}{\Gamma^\sigma}_{\tau\nu}A^\tau -{\Gamma^\mu}_{\sigma\rho,\nu}A^\sigma-{\Gamma^\mu}_{\sigma\nu}{\Gamma^\sigma}_{\tau\rho}A^\tau\\ &=({\Gamma^\mu}_{\tau\nu,\rho}+{\Gamma^\mu}_{\sigma\rho}{\Gamma^\sigma}_{\tau\nu} -{\Gamma^\mu}_{\tau\rho,\nu}-{\Gamma^\mu}_{\sigma\nu}{\Gamma^\sigma}_{\tau\rho})A^\tau\\ &=(\partial_\rho{\Gamma^\mu}_{\nu\tau}-\partial_\nu{\Gamma^\mu}_{\rho\tau}+{\Gamma^\mu}_{\rho\sigma}{\Gamma^\sigma}_{\nu\tau}-{\Gamma^\mu}_{\nu\sigma}{\Gamma^\sigma}_{\rho\tau})A^\tau \end{align}
as desired.