On Chevalley’s formula for structure constants

Suppose g = a semi-simple Lie algebra over C G = the corresponding adjoint group B = a Borel subgroup of G T = a Cartan subgroup of B b, t = corresponding Lie algebras Σ = roots corresponding to the choice of t ∆ = simple roots corresponding to the choice of b W = the Weyl groupNG(T )/T . For each α in ∆, let eα 6= 0 be an element of the root space gα. The triple (b, t, {eα}) is called a frame for g. The set of frames is a principal homogeneous space for G. For each α in∆ there exists a unique e−α in g−α such that tα = −[eα, e−α] lies in t and satisfies the equation 〈α, tα〉 = 2. Let θ be the opposition involution of g (and of G) taking each eα to e−α, acting as −1 on t (and t 7→ t −1 on the torus T ). For example, if g = sl2 then these could be