Let ℒ be a finite-dimensional split semisimple Lie algebra over a field of field characteristic 0, ℋ a splitting Cartan subalgebra, and Λ a weight of ℋ in a representation of ℒ. Then Λ' = Λ S_α = λ - (2(Λ, α))/(α, α)(α) is also a weight. Furthermore, the reflections S_α with α a root, generate a group of linear transformations in ℋ_0^* called the Weyl group W of ℒ relative to ℋ, where ℋ^* is the algebraic conjugate space of ℋ and ℋ_0^* is the Q-space spanned by the roots (Jacobson 1979, pp. 112, 117, and 119).

Cartan matrix | Dynkin diagram | Lie algebra | Lie algebra root | Lie group | Macdonald's constant-term conjecture | root lattice | root system | semisimple Lie algebra