Conjugation of a Weyl element

Let $Φ \subset 𝔼$ be a root system, and $𝐺$ be the group of all invertible linear endomorphisms leaving $Φ$ invariant, i.e.

𝐺 = {𝜑 \in G L (𝔼) : 𝜑 (Φ) = Φ}

Then the Weyl group $W$ of $Φ$ is a normal subgroup of $𝐺$ , in particular for any $𝛼 \in Φ$ and $𝜏 \in 𝐺$ #m/thm/geo

𝜏 𝜎 𝛼 𝜏 - 1 = 𝜎 𝜎 (𝛼)

Therewithal¹

⟨ 𝛽, 𝛼 ⟩ = ⟨ 𝜑 (𝛽), 𝜑 (𝛼) ⟩

Proof

#missing/proof

#state/develop | #lang/en | #SemBr

Footnotes