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  • $\begingroup$ I think this is answering an ever-so-slightly different question than what was asked. The question asks how to derive Maxwell's equations directly from $\mathcal{L} = A_\mu (g^{\mu\nu}\partial^2 - \partial^\mu \partial^\nu)A_\nu$. $\endgroup$ Commented Jan 15 at 6:28
  • $\begingroup$ @user34722 See my edit. $\endgroup$ Commented Jan 15 at 6:54