Lagrange multiplier for antisymmetry ($\partial \mathcal{L}/\partial M^{[\mu \nu ]}=0$)

To vary an antisymmetric tensor without treating its entries as independent physical variables, temporarily allow an unrestricted array $M^{\mu \nu }$ and add ${\lambda }_{\mu \nu }(M^{\mu \nu }+M^{\nu \mu })/2$ to the Lagrangian density, with symmetric Lagrange multiplier $\lambda$. Variation of $\lambda$ enforces antisymmetry. The antisymmetric part of the unrestricted $M$ equation eliminates $\lambda$ and gives the physical Euler-Lagrange field equation. Equivalently, vary only the six independent entries in four spacetime dimensions. Quadratic contractions must be differentiated before applying antisymmetry, or their factors and signs can be lost.