= 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 $\dim Z_{\mathfrak g}(x)\geq3\ell-2$. 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 $\mathbb Cx$, these give $2(\ell-1)+(\ell-1)+1$ independent vectors in the <Lie algebra centralizer>.
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