Let $E$ be a finite-dimensional real Euclidean [vector space](/page/Vector%20Space) with inner product $(\cdot,\cdot)_E$, let $\Phi \subset E$ be a reduced root system, and let $\Delta \subset \Phi$ be a base. For each root $\alpha \in \Phi$, let
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\begin{align*}
s_\alpha: E &\to E \\
x &\mapsto x - \langle x,\alpha^\vee\rangle \alpha
\end{align*}
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be the reflection through the hyperplane orthogonal to $\alpha$, where $\alpha^\vee := 2\alpha/(\alpha,\alpha)_E$ and $\langle x,\alpha^\vee\rangle := 2(x,\alpha)_E/(\alpha,\alpha)_E$. If $W := \langle s_\beta : \beta \in \Phi\rangle$ is the Weyl group of $\Phi$, then
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\begin{align*}
W = \langle s_\alpha : \alpha \in \Delta\rangle.
\end{align*}