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  associated to  a  function $f$  is  denoted  by  $H_f$.

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  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?