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Let $\mathfrak{g}$ be a finite-dimensional real semisimple Lie algebra and $\Sigma$ its root system (not necessarily reduced). Let $\alpha \in \Sigma$. Then $$ \mathfrak{l} := ( \mathfrak{g}_\alpha \oplus \mathfrak{g}_{2\alpha} ) \oplus (\mathfrak{g}_{-\alpha} \oplus \mathfrak{g}_{-2\alpha} ) \oplus ( [\mathfrak{g}_\alpha , \mathfrak{g}_{-\alpha}] \oplus [\mathfrak{g}_{2\alpha}, \mathfrak{g}_{-2\alpha}] ) $$ is a subalgebra. One can prove that it has rank one. Is $\mathfrak{l}$ semisimple? If $\mathfrak{l}$ can be reductive, what whould be an example?

Strichcoder
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    Interesting question. But note that the very last sum should not be direct. Already in the easiest non-trival example, $\mathfrak g = \mathfrak{su}{1,2}$ (spelled out a bit in https://math.stackexchange.com/a/2497093/96384), the nontrivial $[\mathfrak{g}{2\alpha}, \mathfrak{g}{-2\alpha}]$ is actually contained in $[\mathfrak{g}{\alpha}, \mathfrak{g}_{-\alpha}]$. – Torsten Schoeneberg Jan 17 '24 at 21:46

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