Let $\Phi \subset E$ be a finite reduced root system in a finite-dimensional real Euclidean [vector space](/page/Vector%20Space) $E$, and let $W$ be the Weyl group generated by the root reflections $s_\alpha$ for $\alpha \in \Phi$. For each $\alpha \in \Phi$, let
be the reflecting hyperplane. A Weyl chamber is a connected component of
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\begin{align*}
E \setminus \bigcup_{\alpha \in \Phi} H_\alpha .
\end{align*}
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Then the natural action of $W$ on the set of Weyl chambers is simply transitive: for any two Weyl chambers $C$ and $C_1$, there exists a unique element $w \in W$ such that $w(C)=C_1$.