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Let $L$ be a finite dimensional complex semisimple Lie algebra.

Let $H$ be a maximal abelian toral subalgebra (Cartan subalgebra). Let $$L=H\oplus \bigoplus_{\alpha\in \Phi} L_{\alpha}$$ be root space decomposition w.r.t. $H$. My question is simple:

Q. Does $H$ always uniquely determines the components $L_{\alpha}$?

I was thinking that this should be true, $L_{\alpha}$ is largest subspace of $L$ on which $H$ acts (via ad) by scalar mumtiplication by (scalar function) $\alpha$, i.e. $$[h,x]=\alpha(h)x \hskip5mm \mbox{ for all } x\in L_{\alpha}, h\in H.$$ Regarding this action, the space $L_{\alpha}$ is uniquely determined.

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    $\begingroup$ Yes, once $H$ has been chosen then $L_{\alpha}$ is uniquely determined. $\endgroup$ – David Towers May 10 '17 at 8:35
  • $\begingroup$ Once $H$ is fixed the root space decomposition if fixed as well. Off course Cartan subalgebras are not unique. $\endgroup$ – Mathematician 42 May 10 '17 at 9:23

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