Write $\epsilon_{\mu\nu\alpha\beta} F^{\mu\nu} F^{\alpha\beta}$ as a total divergence $\partial_\mu G^\mu$ I have the following homework problem in theoretical electrodynamics:

Show that the gauge invariant Lagrange density $\epsilon_{\mu\nu\alpha\beta} F^{\mu\nu} F^{\alpha\beta}$ can be written as a total divergence of a four-vector.

This total divergence should be something like $\partial_\mu G^\mu$, right?
With
$$
\hat F^{\mu\nu} = \frac 12 \epsilon^{\mu\nu\alpha\beta} F_{\alpha\beta}
$$
I figured that I write this Lagrange density (I'll refer to it as $L_2$) could be written as:
$$
L_2 = 2 \hat F^{\mu\nu} F_{\mu\nu}
$$
I simplified this by using the matrix representations of each $F$ and got it down to:
$$
L_2 = -\frac 4c B_i E^i
$$
$B$ and $E$ can be expressed in Terms of $A$ like so:
$$
B_i = \epsilon_{ijk} \partial_j A^j
,\quad
E_i = -\partial_i A^0 -\partial_0 A^i
$$
How would I continue to find that four-vector $G$?
 A: If you write $$F_{\mu\nu} = \partial_{[\mu}A_{\nu ]} $$ Then your Lagrange density is $${ \mathcal{L}} = \epsilon^{\mu\nu\alpha\beta}\partial_{[\mu}A_{\nu ]}\partial_{[\alpha}A_{\beta ]}$$  
Now we want a vector whose divergence is ${ \mathcal{L}}$.  The $\partial_{\mu}$ looks promising, so we ask if we can bring that outside so it acts on the remaining vector, i.e. $${ \mathcal{L}} = \partial_{\mu}(\epsilon^{\mu\nu\alpha\beta}A_{\nu }\partial_{[\alpha}A_{\beta ]}) \ \ (1)$$  We don't need to worry that we've lost the antisymmetrization brackets on $\mu$ and $\nu$ because the epsilon symbol forces this.
As you pointed out, this isn't quite what we started with because we have an additional term of the form $$(\epsilon^{\mu\nu\alpha\beta}A_{\nu }\partial_{\mu}\partial_{[\alpha}A_{\beta ]}) $$  However the total antisymmetry of the epsilon symbol means we can treat the $\mu$ $\alpha$ $\beta$ contribution as $$\partial_{[\mu}F_{\alpha\beta]} $$ which vanishes due to the Maxwell equations.  Hence the anzatz (1) holds.  (The vector whose divergence we take looks like the abelian Chern Simons current).
