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Suppose $\mathfrak{g}$ is a simple Lie algebra. The standard Cartan-Weyl basis is denoted as $H^I, E^\alpha$ where $\alpha $ denotes the roots, and $E^\alpha$ are the ladder operators. Let $\alpha_i$ be the simple roots.

Suppose we know $\alpha = \sum_{i = 1}^r m_i \alpha_i$. Is there a systematic way to write $E^\alpha $ in terms of nested commutator of $E^{\alpha_i}$, given the knowledge on $m_i$?

Note that some ordering of nested commutator gives zero, namely it's possible that

$$ [[E^{\alpha_i}, E^{\alpha_j}], E^{\alpha_k}] = 0, \quad \text{but} \quad [[E^{\alpha_i}, E^{\alpha_k}], E^{\alpha_j}] \ne 0 \ . $$

Lelouch
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    There is a helpful lemma in root systems that says there is at least one path from some simple root $\alpha_i$, adding one simple root in each step, such that all steps are roots and you end up at your $\alpha$. Say in $G_2$ let $\alpha, \beta$ be the standard basis with $\alpha$ short, then if your desired root is $\beta+2\alpha$, you can get there via $\beta, \beta +\alpha, \beta+2\alpha$. (Or you start with $\alpha$, then $\alpha+\beta$, then $2\alpha +\beta$. But you cannot go through $2\alpha$ because that's not a root.) This gives you at least one way to order your nesting. – Torsten Schoeneberg Apr 21 '23 at 18:15

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