The standard way to write the Maxwell equations (say in vacuum in absence of charges) as Euler-Lagrange (EL) equations is to take the first pair of the Maxwell equaitons and to deduce from it existence of electromagnetic potential. Substituting it into the second pair of Maxwell equations one gets second order equations for the potential. They can be presented as Euler-Lagrange equations for the potential.
I am wondering if there is a way to present the Maxwell equations as an EL-equation in terms of electromagnetic fields only rather than potential.
I think I can prove that this is impossible if one requires in addition that the Lagrangian density is invariant under the Poincare group.