## How to show this $L$-module is simple? related to the root space decomposition for semisimle Lie algebras

Given a semisimple Lie algebra (finite dimensional over a field $$K$$ characteristic $$0$$ and algebraically closed), there exists a root space decomposition $$L = H \oplus \oplus_{\alpha \in R} L_{\alpha},$$ where $$H$$ is a maximal toral subalgebra, $$R = \{\alpha \in H^* : L_{\alpha} \not = 0 , \alpha \not = 0 \}$$ and $$L_{\alpha} = \{ x \in L : ad h(x) = \alpha(h) x \ \forall h \in H \}.$$

I want to prove that $$(L_{\alpha} + L_{-\alpha} + [L_{\alpha},L_{-\alpha}])$$-module $$\sum_{j \in \mathbb{Z}} L_{\beta+ j \alpha}$$ is simple when $$\alpha$$ and $$\beta$$ are linearly independent roots.
I know that each $$L_{\alpha}$$ is one dimensional for $$\alpha \in R$$.

Any comments would be appreciated. Thank you!