Let $\Phi$ be a reduced root system in a finite-dimensional real [inner product space](/page/Inner%20Product%20Space) $(V,(\cdot,\cdot))$. Let $\Delta \subset \Phi$ be a base, and let $\Phi^+$ be the positive system determined by $\Delta$. For each $\alpha \in \Delta$, define the simple reflection
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\begin{align*}
s_\alpha: V &\to V \\
v &\mapsto v - (v,\alpha^\vee)\alpha,
\end{align*}
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where $\alpha^\vee := 2\alpha/(\alpha,\alpha)$. Then, for every $\alpha \in \Delta$,