Root-space reducedness lemma ($\dim {\mathfrak{g}}_{\alpha }=1,R\cap \mathbb{R}\alpha =\{\alpha ,-\alpha \}$)

For a complex semisimple Lie algebra, restrict the module $\mathfrak{h}\oplus {⨁}_{j\ne 0}{\mathfrak{g}}_{j\alpha }$ to the sl2 subalgebra associated with a root. All its weights are even. Its raising operator maps the weight-zero space $\mathfrak{h}$ onto the one-dimensional line of the chosen root vector, so there is exactly one nontrivial irreducible summand: the adjoint module of highest weight $2$. Thus the root spaces at $\pm \alpha$ are one-dimensional and $2\alpha$ is not a root. For any parallel root $c\alpha$, the integers $2c$ and $2/c$ have product $4$; excluding double and half roots leaves only $c=\pm 1$.