Let $\Phi$ be a reduced crystallographic root system in a real Euclidean [vector space](/page/Vector%20Space), let $\Delta = \{\alpha_1,\dots,\alpha_r\}$ be a choice of simple roots, and let $W$ be the Weyl group generated by the simple reflections $s_i := s_{\alpha_i}$ for $1 \leq i \leq r$. For each pair $1 \leq i,j \leq r$, define

\begin{align*} m_{ij} := \operatorname{ord}(s_i s_j) \in \mathbb{N}. \end{align*}

Then $m_{ii}=1$, and the homomorphism from the abstract group

\begin{align*} \left\langle t_1,\dots,t_r \mid (t_i t_j)^{m_{ij}}=e \text{ for all } 1 \leq i,j \leq r \right\rangle \end{align*}

to $W$ determined by $t_i \mapsto s_i$ is an isomorphism. Equivalently,

\begin{align*} W \cong \left\langle s_1,\dots,s_r \mid (s_i s_j)^{m_{ij}}=e \text{ for all } 1 \leq i,j \leq r \right\rangle. \end{align*}