A complex semisimple Lie algebra g with Cartan matrix (cα,β) is the free Lie algebra generated by {eα, fα, hα : α ∈ ∆} modulo the ideal generated by the relations: (1) [hα, hβ] = 0 , [eα, fβ] = δα,βhα (2) [hα, eβ] = cα,βeβ , [hα, fβ] = −cα,βfβ (3) (ad eα)1−cα,β eβ = 0 if α = β (4) (ad fα)1−cα,β fβ = 0 if α = β. Relations (3), (4) are often called the Serre relations. In U(g) they take the equivalent form: (3’) 1−cα,β k=0 1−cα,β k e1−cα,β−k α eβek α = 0 if α = β (4’) 1−cα,β k=0 1−cα,β k f 1−cα,β−k α fβf k α = 0 if α = β.