Is the Lie-subalgebra generated by the root-spaces of a semisimple Lie-algebra semisimple?

Question

Let $\mathfrak{g}$ be a finite-dimensional real semisimple Lie algebra and $\Sigma$ its root system (not necessarily reduced). Let $\alpha \in \Sigma$. Then $$ \mathfrak{l} := ( \mathfrak{g}_\alpha \oplus \mathfrak{g}_{2\alpha} ) \oplus (\mathfrak{g}_{-\alpha} \oplus \mathfrak{g}_{-2\alpha} ) \oplus ( [\mathfrak{g}_\alpha , \mathfrak{g}_{-\alpha}] \oplus [\mathfrak{g}_{2\alpha}, \mathfrak{g}_{-2\alpha}] ) $$ is a subalgebra. One can prove that it has rank one. Is $\mathfrak{l}$ semisimple? If $\mathfrak{l}$ can be reductive, what whould be an example?