Proving the Maxwell energy-momentum tensor is divergence-free

[ClaraOxford]

How do you prove that Maxwell's energy-momentum equation is divergence-free?
I don't know whether or not I have to use Lagrangians or Eistein's tensor, or if there's a simlpler way of expanding out the tensor..

∂$_{\mu}$T$^{\mu\nu}$=0

T$^{}\mu\nu$=F$^{}\mu\alpha$F$^{}\nu$$_{}\alpha$-1/4F$^{}\alpha\beta$F$_{}\alpha\beta$$\eta$$^{}\mu\nu$

 

[ClaraOxford]

I mean

∂$_{\mu}$T$^{\mu\nu}$=0

T$^{\mu\nu}$=F$^{\mu\alpha}$F$^{\nu}$$_{\alpha}$-1/4F$^{\alpha\beta}$F$_{\alpha\beta}$$\eta$$^{\mu\nu}$

 

[vela]

Try writing $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ and using the commutativity of the derivatives.