I have a bit of a contradiction in my brain and I was hoping once again that excellent Mathoverflow community could help me out :)

Let $\rho_g$ be the action associated to a non-abelian Lie Group $G$ on a manifold $M$ and take $\eta,\nu \in \mathfrak{g}$ such that $[\eta,\nu]=\zeta$. It seems that in order for the group action to be effective, the fundamental vector fields $X_{\eta}, X_{\nu}$ associated to $\eta,\nu$ do not commute as given the usual Lie algebra homomorphism $A:\mathfrak{g}\rightarrow \mathfrak{X}(M)$ we have that $[X_{\eta}, X_{\nu}]=A([\eta, \nu])=A(\zeta)\neq 0$ if the action is effective.

On the other hand, if such an action is Hamiltonian with respect to a symplectic form $\omega$ on $M$ we know that the orbits are isotropic submanfolds and so $\omega(X_{\eta}, X_{\nu})=0$. However this implies that the Hamiltonian vector fields $X_{\eta}, X_{\nu}$ commute... Where have I gone wrong?

Edit: I'm fine to assume that $G$ is compact, if this makes everything easier