α_{1}, \dots, α_{r}

. For the orbit statement one must use the Closed dominant Weyl chamber

W = {λ : ⟨ λ, α_{i}^{\lor} ⟩ \geq 0 for every i} .

(1)

Its interior, defined using strict inequalities, is the open Weyl chamber. Weights on reflecting walls, including zero, cannot be moved into that interior, so closure is necessary here.

Choose

ρ

strictly inside the chamber, for example the sum of the fundamental weights. The finite Weyl group orbit of

β

contains an element

λ

maximizing

(λ, ρ)

. If

(λ, α_{i}) < 0

for some simple root, then

(s_{i} λ, ρ) - (λ, ρ) = - \frac{2 ( λ , α _{i} )}{( α _{i} , α _{i} )} (α_{i}, ρ) > 0,

(2)

contradicting the maximum. Thus every simple-root pairing of

λ

is nonnegative, and

λ \in W

. Consequently some $w \in W$ sends every integral weight $β$ into the closed fundamental chamber. The argument in fact works for every vector in the real weight space.