[[Weyl group]]
# Conjugation of a Weyl element

Let $\Phi \subset \mathbb{E}$ be a [[root system]],
and $G$ be the group of all invertible [[Linear endomorphism|linear endomorphisms]] leaving $\Phi$ invariant, i.e.
$$
\begin{align*}
G = \{ \varphi \in \opn{GL}(\mathbb{E}) : \varphi(\Phi) = \Phi \}
\end{align*}
$$
Then the [[Weyl group]] $\mathcal{W}$ of $\Phi$ is a [[normal subgroup]] of $G$,
in particular for any $\alpha \in \Phi$ and $\tau \in G$ #m/thm/geo
$$
\begin{align*}
\tau \sigma_{\alpha} \tau^{-1} = \sigma_{\sigma(\alpha)}
\end{align*}
$$
Therewithal[^1972]
$$
\begin{align*}
\langle \beta,\alpha \rangle = \langle \varphi(\beta),\varphi(\alpha) \rangle 
\end{align*}
$$

> [!missing]- Proof
> #missing/proof

  [^1972]: 1972\. [[Sources/@humphreysIntroductionLieAlgebras1972|Introduction to Lie Algebras and Representation Theory]], §9.2, p. 43

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