Variation identity for the presymplectic current

Differentiate [A1] in direction ${\delta }_1$ with input ${\delta }_2\varphi$ held fixed, and in direction ${\delta }_2$ with input ${\delta }_1\varphi$ held fixed. The total derivatives of the Euler terms are respectively $({\delta }_1{E}_{\varphi })[{\delta }_2\varphi ]+E_{\varphi }[{\delta }_1{\delta }_2\varphi ]$ and $({\delta }_2{E}_{\varphi })[{\delta }_1\varphi ]+E_{\varphi }[{\delta }_2{\delta }_1\varphi ]$ by the definition of ${\delta }_i{E}_{\varphi }$ in [A1]. Subtract the second differentiated identity from the first. The second-variation terms cancel by [A2], and the left-hand sides cancel by commutation of mixed derivatives. Since $\omega ({\delta }_1,{\delta }_2)={\delta }_2\theta ({\delta }_1)-{\delta }_1\theta ({\delta }_2)$, the remaining identity is $0=({\delta }_1{E}_{\varphi })[{\delta }_2\varphi ]-({\delta }_2{E}_{\varphi })[{\delta }_1\varphi ]-d_M\omega ({\delta }_1\varphi ,{\delta }_2\varphi )\mathrm{.}$

Rearranging step 1.1 gives $d_M\omega =({\delta }_1{E}_{\varphi })[{\delta }_2\varphi ]-({\delta }_2{E}_{\varphi })[{\delta }_1\varphi ]$, with the stated sign fixed by the Iyer–Wald ordering in [A1].