Let $f\in \mathbb{C}[x_1,\dots, x_n]$ 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 $\langle \sigma_1, \dots, \sigma_s, \delta_1, \dots, \delta_r\rangle$ such that:

- $\sigma_1, \dots, \sigma_s$ are simultaneously diagonalizable,
- $\delta_1, \dots, \delta_r$ are nilpotent,
- $[\sigma_i,\delta_j]\in \mathbb{Q} \cdot \delta_j$ for all i,j.

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