I have been asked to express the curvature tensor in terms of the metric tensor, with $\mathcal{R}$ being the Riemann curvature tensor, given in terms of the Christoffel symbols of the second kind $${\mathcal R}{^\rho_{\sigma\mu\nu}}=\partial_\mu\Gamma^\rho_{\nu\sigma}-\partial_\nu\Gamma^\rho_{\mu\sigma}+\Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma}-\Gamma^\rho_{\nu\lambda}\Gamma^{\lambda}_{\mu\sigma}$$ and this at the same time given as $$\Gamma^\mu_{\nu\lambda}=\frac{1}{2}g^{\mu\kappa}\left[ \partial_\nu g_{\lambda\kappa} +\partial_\lambda g_{\nu\kappa} -\partial_\kappa g_{\nu\lambda}\right]$$ I have found this series of formulas in wikipedia, it starts from the top form of the curvature tensor I provided and then goes on to lower the top index with $g_{\gamma\rho}$ and treats it as if it were straighforward. However, I have, per example, expanded the two terms with first partial derivatives, like: \begin{aligned} \partial_\mu\Gamma^\rho_{\nu\sigma}&=\frac{1}{2}\partial_\mu\left[ g^{\mu\kappa}\partial_\nu g_{\lambda\kappa}+g^{\mu\kappa}\partial_\lambda g_{\nu\kappa}-g^{\mu\kappa}\partial_\kappa g_{\nu\lambda} \right]\\ & \begin{split}=\frac{1}{2}&\left[ \partial_\mu g^{\mu\kappa}\partial_\nu g_{\lambda\kappa}+g^{\mu\kappa}\partial_\mu\partial_\nu g_{\lambda\kappa} +\partial_\mu g^{\mu\kappa}\partial_\lambda g_{\nu\kappa}\right.\\ &\left.+g^{\mu\kappa}\partial_\mu\partial_\lambda g_{\nu\kappa}-\partial_\mu g^{\mu\kappa}\partial_\kappa g_{\nu\lambda}-g^{\mu\kappa}\partial_\mu\partial_\kappa g_{\nu\lambda} \right] \end{split} \end{aligned} yet from taking together the two expansions, only 6 terms with second partial derivatives show up (no second partials come from the Christoffel symbols), as opposed to the 8 that should show up. How can I prove the Wikipedia identity? Can I get the lowering metric tensor inside the derivative (as in $\partial_\mu g_{\lambda\rho}\Gamma^{\rho}_{\nu\sigma}$), if so, what would I get? I've never seen a Christoffel symbol contracted with a metric tensor, since once getting the new tensor inside the symbol expansion, I have the same question, can I get the metric tensor inside the partial?
Any help will be greatly appreciated.