I recently started learning about Lie algebras. I just noticed that being a direct sum of Lie algebras is not the same as being a direct sum of vector spaces.
Now If we consider a lie algebra $L$ with a Cartan subalgebra $H$, then there is a Cartan decomposition
\begin{equation} L = H\oplus\bigoplus_{\alpha \in \Phi}L_{\alpha} \end{equation} So my question is pretty simple and it is just to be sure. Here we have a direct sum of vector spaces and not lie algebras right? for example $[L_{\alpha},L_{-\alpha}]\subset H$ is different than zero in general, no?