Just to add a little detail to the discussion above and since what I wrote was not clear:

$E(M,\omega)$ is, in Lichnerowicz-Avez-Diaz Miranda (I forgot the third author in the above citation), the Lie algebra of infinitesimal conformal symplectic transformations (Paragraph 5).

$Z(M,\omega)$ is, in Lichnerowicz-Avez, the Lie algebra of symplectic vector fields (equivalently locally hamiltonian vector fields).

$Z^\prime(M,\omega)$ is the normalizer (in Lie algebras you use this term rather than idealizer) of $Z(M,\omega)$ inside $E(M,\omega)$, i.e. the set $\{X\in E(M,\omega)\, : [X,Y]\in Z(M,\omega)\,\forall Y\in Z(M,\omega)\}$.

At page 12 it is shown that $[E(M,\omega),E(M,\omega)]\subseteq Z(M,\omega)$; therefore $Z^\prime(M,\omega)=E(M,\omega)$, if I am not wrong and/or confused by different terminology and notations.

(btw Proposition 2 of the mentioned paper may be of interest to you)

Just to add a little detail to the discussion above and since what I wrote was not clear:

$E(M,\omega)$ is, in Lichnerowicz-Avez-Diaz Miranda (I forgot the third author in the above citation), the Lie algebra of infinitesimal conformal symplectic transformations (Paragraph 5).

$Z(M,\omega)$ is, in Lichnerowicz-Avez, the Lie algebra of symplectic vector fields (equivalently locally hamiltonian vector fields).

$Z^\prime(M,\omega)$ is the normalizer (in Lie algebras you use this term rather than idealizer) of $Z(M,\omega)$ inside $E(M,\omega)$, i.e. the set $\{X\in E(M,\omega)\, : [X,Y]\in Z(M,\omega)\,\forall Y\in Z(M,\omega)\}$.

At page 12 it is shown that $[E(M,\omega),E(M,\omega)]\subseteq Z(M,\omega)$; therefore $Z^\prime(M,\omega)=E(M,\omega)$, if I am not wrong and/or confused by different terminology and notations.

(btw Proposition 2 of the mentioned paper may be of interest to you)