I'm going crazy! (Electromagnetic Energy-Momentum Tensor)

[LineIntegral]

1. The problem statement, all variables and given/known data

The problem is conveniently located here:

http://www.dur.ac.uk/resources/cpt/graduate/lectures/mscps.pdf

Problem no. 31. There's even a solution, here:

http://www.dur.ac.uk/resources/cpt/graduate/lectures/grsolns.pdf

However, I don't understand the solution. How does the antisymmetry of F^{\rho\lambda} allow us to rewrite \nabla_\rho F_{\sigma\lambda} - \frac{1}{2} \nabla_\sigma F_{\rho\lambda} as \frac{1}{2}\left ( \nabla_\rho F_{\sigma\lambda} - \nabla_\lambda F_{\sigma\rho} - \nabla_\sigma F_{\rho\lambda} \right ) ? I just don't see it. This implies that \nabla_\rho F_{\sigma\lambda} = -\nabla_\lambda F_{\sigma\rho}, which doesn't seem right...

2. Relevant equations

See above.


3. The attempt at a solution

See above.

Any help will be much appreciated!

[fzero]

The point is that you have

F^{\rho\lambda} \nabla_\rho F_{\sigma\lambda} = \frac{1}{2} (F^{\rho\lambda}-F^{\lambda\rho}) \nabla_\rho F_{\sigma\lambda}.

After a relabeling of dummy indices, you find the claimed result.

[LineIntegral]

Thanks! Now it finally makes sense... :)