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Let $\delta$ be a derivation of a complex Lie algebra L, and for $\lambda \in C$, let $$L_{\lambda}=\lbrace x \in L:(\delta-\lambda 1_{L})^{m}\;x=0 \mbox{ for some } m \ge 1 \rbrace$$
be the generalised eigenspace of $\delta$ corresponding to $\lambda$.

Why $[L_{\lambda},L_{\mu}] \subseteq L_{\lambda+\mu}$?

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  • $\begingroup$ Why $[L_{\lambda},L_{\mu}] \subseteq L_{\lambda+\mu}?$ $\endgroup$ Jul 20, 2019 at 23:30
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    $\begingroup$ See e.g. the formula in Lemma 5.6 in my lectures here, which implies that if $x\in\mathrm{Ker}((\delta-\lambda 1)^k)$ and $y\in\mathrm{Ker}((\delta-\mu 1)^\ell)$ then $[x,y]\in\mathrm{Ker}((\delta-(\lambda+\mu) 1)^{k+\ell-1})$. $\endgroup$
    – YCor
    Jul 20, 2019 at 23:45

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