Let $G$ be a connected, simply-connected complex semisimple Lie group with Lie algebra $\frak{g}$. Fix a maximal torus $T\subseteq G$, and let $$\frak{g}=\frak{t}\oplus\bigoplus_{\alpha\in\Delta}\frak{g}_{\alpha}$$ be the corresponding decomposition into weight spaces. Given a non-zero root vector $e_{\alpha}\in\frak{g}_{\alpha}$, is there a nice way to describe those root vectors lying in the nilpotent $G$-orbit of $e_{\alpha}$? This is possible in the context of several examples, but is there a more example-independent description?
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