= Weyl orbit spans an irreducible root space
{c}
{title2=$\operatorname{span}(W\alpha)=V$}
For an <irreducible root system> spanning $V$, the span $U$ of a nonzero root's <Weyl group> orbit is invariant, as is $U^\perp$. A root reflection preserving both subspaces must have its normal entirely in one of them: otherwise its rank-one correction mixes the subspaces. Thus a proper nonzero $U$ would partition all roots into mutually orthogonal subsets, contradicting irreducibility. Consequently some orbit root is nonorthogonal to any given root. Since <Weyl reflections> preserve lengths, the pairwise integrality restrictions for nonorthogonal roots also bound length ratios of orthogonal roots. All squared-length ratios lie in $\{1,2,1/2,3,1/3\}$; there cannot be three distinct lengths, since ratios two and three relative to a shortest root would yield the forbidden ratio $3/2$.
Back to article page
