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!