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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, Xα for 𝔤, where Hi, i =1, 2, ... , r is a basis for a Cartan subalgebra, and where Xα, Xα, 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, Xα = 1 and set Hα = Xα,Xα. Scale the vectors Xα again (preserving BXα, Xα = 1 ) so that the structure equations

 Hα,Xα = 2 Xα,   Xα,Xα=Hα ,  Hα,Xα = 2 Xα

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, Xα 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 Xαi 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.