= 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 $\mathbb Cx$ also centralizes $x$, and $[x,\mathfrak g_\theta]\subseteq\mathfrak g_{\alpha+\theta}=0$. These independent spaces give
$$
\dim\mathfrak z_{\mathfrak g}(x)\geq(\ell-1)+1+1=\ell+1.
$$
Hence \b[a nonzero simple-root vector is not regular when $\ell>1$].
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