I was wondering why we need the electromagnetic field tensor $F_{\mu\nu}$ to be a tensor and why can't we work with the electric and magnetic fields while dealing with the electromagnetic field Lagrangian density for flat spacetime. I kind of get the intuition behind it, because you cannot write the transformations as vector equations and need a second-order tensor, but where exactly does the math go wrong? I would prefer a purely mathematical reasoning, if possible.