= Root-space reducedness lemma
{title2=$\dim\mathfrak g_\alpha=1,\quad R\cap\mathbb R\alpha=\{\alpha,-\alpha\}$}
For a complex <semisimple Lie algebra>, restrict the module $\mathfrak h\oplus\bigoplus_{j\ne0}\mathfrak g_{j\alpha}$ 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 $c=\pm1$.
Back to article page
