Let  $(M,\omega)$  be a $2n$ dimensional symplectic  manifold and  $X$ is  a smooth vector  field on $M$. Consider the following subvector  space of $\chi^{\infty}(M)$: $$S(X)=\{Y\in \chi^{\infty}|(M)\mid  X.\omega(X,Y)=nDiv(X)\omega(X,Y)\}$$.


This  vector  space  contains the Lie algebra $C(X)=\{Y\in \chi^{\infty}(M)\mid [X,Y]=0\}$.  It also contains the  Lie  algebra $M(X)=\{fX\mid f\in C^{\infty}(M)\}$

>Is $S(X)$ a Li subalgebra of $\chi^{\infty}(M)$?

> If the  answer is no, is the Lie  algebra  generated by $S(X)$ equal to the lie  algebra generated by $C(X)$  and $M(X)$?