I am trying to prove that symplectic vector fields with the usual Lie bracket of vector fields form a Lie Algebra. Recall that a vector field $X$ on a symplectic manifold $(M,\omega)$ is symplectic iff $\mathcal{L}_{X}\omega = 0$.
I can prove skew-symmetry, since by cartan homotopy formula $\mathcal{L}_{[X,Y]}\omega = d \mathcal{i}_{[X,Y]}\omega$, which equals $-d \mathcal{i}_{[Y,X]} \omega$ by skew-symmetry of $[\cdot,\cdot]$ and of $\mathcal{i}$, and this equals $-\mathcal{L}_{[Y,X]}\omega$.
I would like to ask how to prove Jacobi identity. Any tips would be appreciated.