Let $L$ be a semisimple Lie algebra, ad let $\Phi$ be a root system. Fix a fundamental root system $\Delta$ of $\Phi$ with corresponding to $\Phi^+$.
I would like to understand the subalgebra generated by all $L_{\alpha}$ and $L_{-\alpha}$ where $\alpha\in \Omega\subset\Delta$. Is this subalgebra semisipmle? Can we say something about its Dynkin diagram?