The Weyl group of a semi-simple Lie group, a semi-simple Lie algebra, a semi-simple linear algebraic group, etc. is the Weyl group of the root system of that group or algebra.

Removing the hyperplanes defined by the roots of Φ cuts up Euclidean space into a finite number of open regions, called **Weyl chambers**. These are permuted by the action of the Weyl group, and it is a theorem that this action is simply transitive. In particular, the number of Weyl chambers equals the order of the Weyl group. Any non-zero vector *v* divides the Euclidean space into two half-spaces bounding *v*^{∧}, namely *v*^{+} and *v*^{−}. If *v* belongs to some Weyl chamber, no root lies in *v*^{∧}, so every root lies in *v*^{+} or *v*^{−}, and if α lies in one then −α lies in the other. Thus Φ^{+} := Φ∩*v*^{+} consists of exactly half of the roots of Φ. Of course, Φ^{+} depends on *v*, but it does not change if *v* stays in the same Weyl chamber. The base of the root system with respect to the choice Φ is the set of *simple roots* in Φ^{+}, i.e., roots which cannot be written as a sum of two roots in Φ^{+}. Thus, the Weyl chamber, the set &Phi^{+}, and the base determine one another, and the Weyl group acts simply transitively in each case. The following illustration shows the six Weyl chambers of the root system A_{2}, a choice of *v*, the hyperplane *v*^{∧} (indicated by a dotted line), and positive roots α, β, and γ. The base in this case is {α,γ}.

Weyl groups are examples of Coxeter groups. This means that they have a special kind of presentation in which each generator *x _{i}* is of order two, and the relations other than

If *G* is a semisimple linear algebraic group over an algebraically closed field (more generally a *split* group), and *T* is a
maximal torus, the normalizer *N* of *T* contains *T* as a subgroup of finite index, and the Weyl group *W* of *G* is isomorphic to *N/G*. If *B* is a Borel subgroup of *G*, i.e., a maximal connected solvable subgroup and *T* is chosen to lie in *B*, then we obtain the **Bruhat decomposition**

which gives rise to the decomposition of the flag variety *G/B* into **Schubert cells** (see Grassmannian).