Let $(M,\omega, g)$ be a Riemannian symplectic manifold. The Poisson bracket of two functions $f,g$ is denoted by $\{f,g\}$. The Hamiltonian vector field and gradient vector field associated to a function $f$ is denoted by $H_f$ and $\nabla_f$, respectively.
We define a Lie algebra structure on $\chi^{\infty}(M)$ as follows:
$$[X,Y]=H_{\{div(X),div(Y)\} }$$
Question 1: Is the Lie algebra $\chi^{\infty}(M)/I$ a simple Lie algebra where $I$ is the ideal $$I=\{X\in \chi^{\infty}(M)\mid div(X)=0\}$$
As the second step for this question we would like to change the definition of the Lie algebra operation by replacing the "Hamiltonian vector field" with "Gradient vector field". But we are not sure that it satisfies the Jackobi identity. So we ask the following question:
Question 2: Under which compatibilty condition between $\omega$ and $g$, the following operation is a Lie algebra bracket on $\chi^{\infty}(M)$:
$$[X,Y]=\nabla_{\{div(X),div(Y)\}}$$ is there an example of this situation with satisfication of the Jackobi identity?
