Centralizer lower bound for a root vector

In a complex semisimple Lie algebra whose rank of a semisimple Lie algebra is $\ell$, a root vector $x$ satisfies $\dim Z_{\mathfrak{g}}(x)\ge 3\ell -2$. To see this, select $\ell -1$ roots whose images form a basis of the real root span modulo $\mathbb{R}\alpha$, where $x\in {\mathfrak{g}}_{\alpha }$. Take the highest endpoints of the root strings through both signs of each selected root. Their $2(\ell -1)$ distinct root spaces commute with $x$. Together with $\ker \alpha \subset \mathfrak{t}$ and $\mathbb{C}x$, these give $2(\ell -1)+(\ell -1)+1$ independent vectors in the Lie algebra centralizer.