I have an issue with Eq. 3.3.6 of Wald's General Relativity. There he would like to prove that for Gaussian normal coordinates, the geodesic tangent field remains orthogonal to all coordinate basis fields which generate hypersurfaces indexed by parameter $t$ (page 43).
The step is as follows: Let $n^a$ be tangent vector to the geodesic and $X^a$ be any one of the coordinate basis fields. Then \begin{equation} \begin{aligned} n^b\nabla_b(n_aX^a)&=n_an^b\nabla_bX^a\\&=n_aX^b\nabla_bn^a\\&=\frac{1}{2}X^b\nabla_b(n^an_a)\\&=0 \end{aligned} \end{equation} First equality is due to product rule and also the definition of geodesics (parallel transport its own tangent vector). Third equality follows by reversing product rule. Fourth equality follows because $n_an^a$ is a constant (norm of $n^a$). But I do not quite get second equality. Wald mentioned that it follows because as coordinate basis for the tangent space at $p$, $[n^a,X^b]=0$. But I cannot see how that enters the second equality.
