Is the Sum of Christoffel Symbols Equal to Their Negative in Tensor Calculus?

[elfmotat]

Is the following true?
$$\Gamma_{\mu \nu \alpha}+\Gamma_{\nu \mu \alpha}=-2\Gamma_{\alpha \mu \nu}$$
where:
$$\Gamma_{\alpha \mu \nu}=g_{\alpha \sigma}\Gamma^{\sigma}_{~\mu \nu}$$

I ask because, while bored in a philosophy lecture, I decided to try to derive the geodesic equation by extremizing ∫g_μνu^μu^νdλ, where u^μ = dx^μ/dλ.

I was able to arrive at the following, where a_μ=du_μ/dλ:
$$2a_\alpha = (\Gamma_{\mu \nu \alpha}+\Gamma_{\nu \mu \alpha})u^\mu u^\nu$$

So, am I on the right track or did I make an error somewhere?

 

[elfmotat]

Nevermind, they are clearly not equal. From the definition of the Christoffel symbols in terms of the metric, I found that:

$$(\Gamma_{\mu \nu \alpha}+\Gamma_{\nu \mu \alpha})=\partial_\alpha g_{\mu \nu }$$
This makes sense, because $\nabla_\alpha g_{\mu \nu }=0$.

Unfortunately for me though, this is clearly not equal to $-2\Gamma_{\alpha \mu \nu}$ given that:

$$-2\Gamma_{\alpha \mu \nu}=\partial_\alpha g_{\mu \nu}-\partial_\mu g_{\nu \alpha}-\partial_\nu g_{\mu \alpha }$$