I am working on the following assignment in Symplectic geometry: Let $G$ be a Lie group acting on a symplectic manifold $(M,\omega)$, and let $X_M=\dfrac{d}{dt}\bigg|_{t=0}\exp(tX)\cdot x$ for $X\in\mathfrak{g}$ be the fundamental vector field. Then the Association $\mathfrak{g}\ni X\mapsto X_M\in\text{SympVec}(M,\omega)$ is an anti-Lie algebra homomorphism.
We have that $$ [X,Y]_M(x)=\frac{d}{dt}\bigg|_{t=0}\exp(t[X,Y])\cdot x=\frac{d}{dt}\bigg|_{t=0}\exp(t\cdot\frac{d}{ds}\bigg|_{s=0}\text{Ad}_{\exp(s X)}Y)\cdot x, $$ How is this equal to $-[X_M,Y_M](x)$?