I'm trying to understand an argument in "An introduction to general relativity" by Hughston and Todd (p37). Let $F_{ab}$ be the electromagnetic field tensor, I'm trying to show:
$$\Box F_{ab} = -4 \pi (\nabla_a J_b - \nabla_b J_a)$$
Now in the text they say that this follows as a consequence of contracting $\nabla_{[a}F_{bc]}$ with $\nabla^a$ however, I'm struggling to see how we could deal with terms of the form $\nabla^a \nabla_b F_{bc}$, I'd like to be able to use the other Maxwell equation $\nabla^a F_{ab}$ however this isn't immediately applicable without switching the order of the derivatives.
I could plausibly do this by switching the order of the derivatives and inserting a Riemann tensor however at this point of the book the Riemann tensor has not been introduced so I wondered if I am overlooking something?