rt.representation theory - Centre of a Lie algebra

Let $fin mathbb{C}[x_1,dots, x_n]$ be a reduced homogeneous polynomial of degree n.

Let $mathfrak{g}={ delta in Der_{mathbb{C}^n}|delta(f)in (f)mathcal{O}_{mathbb{C}^n} text{ and weight}(delta) =0 }$ be a (reductive) complex Lie algebra with minimal system of generators $langle sigma_1, dots, sigma_s, delta_1, dots, delta_rrangle$ such that:

1. $sigma_1, dots, sigma_s$ are simultaneously diagonalizable,


2. $delta_1, dots, delta_r$ are nilpotent,


3. $[sigma_i,delta_j]in mathbb{Q} cdot delta_j$ for all i,j.

Is it true that the centre of $mathfrak{g}$ is made only of diagonalizable elements?