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.
