Chevalley Basis Details
The details for the construction of the Chevalley basis are as follows. Let 𝔤 be a real, split semi-simple Lie algebra. Start with a basis Hi , Xα, X−α for 𝔤, where Hi, i =1, 2, ... , r is a basis for a Cartan subalgebra, and where Xα, X−α, α ∈ Δ+(the positive roots), gives a root space decomposition for 𝔤. By definition of a split, semi-simple Lie algebra, the root vectors are all real. Let B be the Killing form. Scale the vectors Xα such that BXα, X−α = 1 and set Hα = Xα,X−α. Scale the vectors Xα again (preserving BXα, X−α = 1 ) so that the structure equations
Hα,Xα = 2 Xα, Xα,X−α=−Hα , Hα,X−α = −2 X−α
hold. Let Δ0 = α1, α2 , ... , αr be the simple roots, and set
hi = Hαi, xi =Xαi , yi = Yαi, i = 1,2, ... ,r .
This fixes the 3 r vectors h1, h2 , ... , hr , x1, x2, ... , xr, y1, y2, ... , yr in the Chevalley basis ℬ. Write
Δc = Δ+ − Δ0= αr+1, αr+2, ..., α𝓁 .
We need to make one final scaling of the vectors Xα, X−α for α ∈Δc. We calculate the structure constants Xα , Xβ = Nα β Xα + β , for α, β and α + β ∈ Δ+ and generate the system of quadratic equations
q+12tα tβ = Nα β 2 tα +β .
Here q is the largest positive integer such that α −q β is not a root. Put tα = 1 for α ∈ Δ0 and solve for the remaining tα, α ∈ Δc. Finally set uα = tα and put
xi= uαi Xαi and yi = 1/uαi X−αi for i = r+1, r+2, ... 𝓁.
This completes the construction of the Chevalley basis ℬ' h1, h2 , ... , hr , x1, x2, ... , xℓ, y1, y2, ... , yℓ. We have
hi, hj = 0, hi,xj = aijxj , hi,yj = −aijxj
for all i,j = 1,2, ... r, where the matrix aij is the Cartan matrix for 𝔤 and, also,
xi , xj = ±q+1xk where αi + αj = αk.
Note that in the Chevalley basis all the structure constants are integers and that the transformation hi →−hi , xi →yi, yi → xi is a Lie algebra automorphism.
See N. Bourbaki, Lie Groups and Lie Algebras, Chapters 7-9, Section 4 for additional details.
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