How does one see that
$$c g_{ab,cd}\left(\eta^{ac}\eta^{bd} - \eta^{ab}\eta^{cd}\right)$$
is equal to
$$(c/2)\eta^{bc}\eta^{ae}\partial_{a}\left(g_{be,c} + g_{ce,b} - g_{bc,e}\right) - (c/2)\eta^{ae}\eta^{bc}\partial_{c}\left(g_{be,a} + g_{ae,b} - g_{ba,e}\right)$$
where $\eta$ is the Minkowski metric $(+,-,-,-)$ and $g_{ab,cd} = \frac{\partial ^2 g_{ab}}{\partial x^d \partial x^c}$, I keep getting lost :(
First throw away the $c$ since it multiplies both expressions, then distribute the derivatives (i.e. $\partial_a g_{be,c} = g_{be,ca}$). This gives you six scalar terms of the form $\eta^{..} \eta^{..} g_{..,..}$, and due to symmetries there are only two unique such terms: $\eta^{ab} \eta^{cd} g_{ab,cd}$ and $\eta^{ac}\eta^{bd} g_{ab,cd}$. You just need to relabel dummy indices to get each term into one of these forms and then count the signs.
