Weyl Group Is Generated by Root Reflections
Let \(G\) be a connected reductive group with maximal torus \(T\), root system \(\Phi\), and Weyl group \(W=N_G(T)/C_G(T)\). Then \(W\) is generated by the root reflections \(s_\alpha\) for \(\alpha\in\Phi\).
After choosing a Borel subgroup \(B\supseteq T\), the corresponding simple roots \(\Delta\subseteq\Phi\) form a basis of the root system, and \((W,\{s_\alpha:\alpha\in\Delta\})\) is the Coxeter system attached to \((G,T,B)\).
Proof
Each root \(\alpha\) determines a rank-one subgroup \(G_\alpha\) generated by the positive and negative root directions. Its normalizer of \(T\) contains an element acting on \(X^*(T)\) by the reflection \(s_\alpha\), so all root reflections lie in the Weyl group.
The Bruhat and Borel geometry show that any element of \(N_G(T)\) can be written as a product of representatives coming from rank-one root subgroups: successively move a Borel subgroup across codimension-one walls in the chamber decomposition. These wall crossings are exactly the simple reflections. Hence the simple reflections generate \(W\), and all root reflections are their conjugates.