Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 102 / 3 / b / Solution

Choose a positive system of a root system in which $\alpha$ is a simple root; this is possible after applying an element of the Weyl group. Let $\theta$ be the highest root. Since the rank is greater than one, $\theta \ne \alpha$, and the maximality of $\theta$ implies that $\theta +\alpha$ is not a root.

For $0\ne x\in {\mathfrak{g}}_{\alpha }$, the $(\ell -1)$-dimensional space $\ker \alpha \subset \mathfrak{t}$ centralizes $x$. The line $\mathbb{C}x$ also centralizes $x$, and $[x,{\mathfrak{g}}_{\theta }]\subseteq {\mathfrak{g}}_{\alpha +\theta }=0$. These independent spaces giveHence a nonzero simple-root vector is not regular when $\ell >1$.