Exchange of two covariant derivatives of a 1-form

Question

I'm trying to prove formula $(1.1.3)$ shown below (I've already proven all the ones above) :

I managed to show that:

$$ \nabla_{i} \nabla_{j} v_{k} = \frac{\partial^2 v^k}{\partial x^i \partial x^{j}} - \sum_s \Gamma_{jk}^{s} \frac{\partial v^s}{\partial x^{i}} - \sum_s \frac{\partial \Gamma_{jk}^s}{\partial x^{i}} v_s - \sum_p \Gamma_{ik}^{p} \frac{\partial v_p}{\partial x^s} + \sum_{p, l} \Gamma_{ik}^{p} \Gamma_{jp}^{l}v_l$$

So $\nabla_{i} \nabla_{j} v_{k}$ would be the same, just renaming $i$ to $j$. But I'm not seeing where $g$ is gonna show up here.

If more context is needed, I got it from here, page $9$.

Answer 1

Never mind. It boils down to the fact that $R_{ijk}^m = g^{lm} R_{ijkl}$. It's an easy computation.