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The reflection representation is the action of the Weyl group on the real span of its root system, generated by its Weyl reflections. For an irreducible root system, it is an irreducible representation: if $U$ is an invariant subspace, each root either lies in $U^{\perp }$ or, by subtracting a reflected vector of $U$, lies in $U$. These alternatives partition the roots into orthogonal sets. Irreducibility and spanning force $U=0$ or the whole space.

The Weyl group is generated by the reflections in the roots.

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 $3/2$.