In equation 8 in this paper they claim that in the geodesic equations this can be done:
$ \ddot{x}^\mu = \Gamma^\mu_{\alpha\beta} \dot{x}^\alpha \dot{x}^\beta = - \frac{1}{2} g^{\mu\nu}\left( \partial_\alpha g_{\nu\beta} + \partial_\beta g_{\nu\alpha} - \partial_\nu g_{\alpha\beta} \right) \dot{x}^\alpha \dot{x}^\beta = - g^{\mu\nu}\partial_\beta g_{\nu\alpha} \dot{x}^\alpha \dot{x}^\beta + \frac{1}{2} g^{\mu\nu} \partial_\nu g_{\alpha\beta} \dot{x}^\alpha \dot{x}^\beta $
They claim the later step can be done because:
We can combine the first two terms in equation (8) because the product of an anti-symmetric tensor and a symmetric tensor vanishes—there is no need to explicitly symmetrize $\alpha$ and $\beta$ for $\partial_\beta g_{\nu \alpha} $ from a computational point of view.
I am assuming that the "symmetric tensor" they speak about is $\dot{x}^\alpha \dot{x}^\beta$ but I do not understand the reasoning behind this step. Can anyone further explain in detail why this can be done?
$$\partial_\alpha g_{\nu\beta}\dot{x}^\alpha \dot{x}^\beta = \partial_\alpha g_{\nu\beta}\dot{x}^\beta \dot{x}^\alpha $$ At this point I'm going to be overly pedantic and introduce a different dummy index variable $\gamma$. Since $\beta$ is a summed dummy index we could as well have called it $\gamma$.
$$ \partial_\alpha g_{\nu\beta}\dot{x}^\alpha \dot{x}^\beta = \partial_\alpha g_{\nu\beta}\dot{x}^\beta \dot{x}^\alpha = \partial_\alpha g_{\nu\gamma}\dot{x}^\gamma \dot{x}^\alpha $$ In the last expression, since $\alpha$ is a summed dummy index we could as well have called it $\beta$. $$ \partial_\alpha g_{\nu\beta}\dot{x}^\alpha \dot{x}^\beta = \partial_\alpha g_{\nu\beta}\dot{x}^\beta \dot{x}^\alpha = \partial_\alpha g_{\nu\gamma}\dot{x}^\gamma \dot{x}^\alpha = \partial_\beta g_{\nu\gamma}\dot{x}^\gamma \dot{x}^\beta $$ Finally, in the new last expression, since $\gamma$ is a summed dummy index we could as well have called it $\alpha$. $$ \partial_\alpha g_{\nu\beta}\dot{x}^\alpha \dot{x}^\beta = \partial_\alpha g_{\nu\beta}\dot{x}^\beta \dot{x}^\alpha = \partial_\alpha g_{\nu\gamma}\dot{x}^\gamma \dot{x}^\alpha = \partial_\beta g_{\nu\gamma}\dot{x}^\gamma \dot{x}^\beta= \partial_\beta g_{\nu\alpha}\dot{x}^\alpha \dot{x}^\beta $$ So $$ \frac12\left( \partial_\alpha g_{\nu\beta}\dot{x}^\alpha \dot{x}^\beta +\partial_\beta g_{\nu\alpha}\dot{x}^\alpha \dot{x}^\beta\right) = \frac12\left( 2\partial_\beta g_{\nu\alpha}\dot{x}^\alpha \dot{x}^\beta\right) = \partial_\beta g_{\nu\alpha}\dot{x}^\alpha \dot{x}^\beta $$ as used in the book equation.
After a time or two of working through this sort of expression in detail, it will become natural to just see this sort of thing, and authors tend to gloss over this level of pedantic detail.