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In multiple questions (e.g. A helpful proof in contracting the Christoffel symbol? or https://physics.stackexchange.com/a/101677/290999), I have now seen the following identity being used: $${\Gamma^\mu}_{\mu\lambda}=\frac{1}{2}g^{\mu\rho}(\partial_\mu g_{\lambda\rho}+\partial_\lambda g_{\mu\rho} -\partial_\rho g_{\lambda\mu}) \overset{(*)}{=} \frac{1}{2}g^{\mu\rho}\partial_\lambda g_{\mu\rho}.$$

I just cannot figure out how the first and last term cancel each other out in the second term, i.e. why the equation I marked $(*)$ works out. As I understood it, the term in the middle (as well as the one on the right hand side) has an implied summation over $\rho$. I cannot see how the $\partial_\rho g_{\lambda\mu}$-terms vanish for $\rho\neq\mu$.

I've tried it multiple times and in my calculations it always comes out as wrong.

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Since $g^{\mu\rho}\partial_\mu g_{\lambda\rho}=g^{\rho\mu}\partial_\mu g_{\lambda\rho}=g^{\mu\rho}\partial_\rho g_{\lambda\mu}$ (first use $g$'s symmetry then exchange two index labels), the other terms cancel.

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