Denote by $\mathfrak{g}$ a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. Let $\Phi$ be the root system of $(\mathfrak{g},\mathfrak{h})$ and denote by $\mathfrak{g}_\alpha$ the root subspace of $\mathfrak{g}$ corresponding to a root $\alpha$. We fix a choice of positive roots $\Phi^+$, and let $\Delta$ be the corresponding subset of simple roots in $\Phi^+$. Note that each subset $I\subseteq\Delta$ generates a root system $\Phi_I\subseteq\Phi$, with positive roots $\Phi_I^+:=\Phi_I\cap \Phi^+$.
Let $ \mathfrak{l}_I:=\mathfrak{h}\oplus\sum_{\alpha\in\Phi_I}\mathfrak{g}_\alpha $.
My questions:
Is $\mathfrak{l}_I$ complex semsimple? If so, why?
Is $\mathfrak{l}_I$ a Levi subalgebra of $\mathfrak{g}$?