First-order Maxwell action ($S=\frac{1}{16\pi }\int \sqrt{-g}[F^{ab}{F}_{ab}-2F^{ab}({\nabla }_a{A}_b-{\nabla }_b{A}_a)]$)

Vary the antisymmetric auxiliary electromagnetic field tensor and the electromagnetic four-potential independently. The first variation sets $F_{ab}={\nabla }_a{A}_b-{\nabla }_b{A}_a$ algebraically, and variation of the potential gives ${\nabla }_b{F}^{ab}=0$. Substitution of the algebraic equation reduces the first-order action to the usual negative quadratic Maxwell action. Metric variation gives the same stress tensor after eliminating the auxiliary field, because the chain-rule term multiplying its variational equation then vanishes.