 ChevalleyBasis Details - Maple Help

 Chevalley Basis Details   The details for the construction of the Chevalley basis are as follows. Let be a real, split semi-simple Lie algebra. Start with a basis for $𝔤$, where ${H}_{i}$, is a basis for a Cartan subalgebra, and where , (the positive roots), gives a root space decomposition for $\mathrm{𝔤}$. By definition of a split, semi-simple Lie algebra, the root vectors are all real. Let $B$ be the Killing form. Scale the vectorssuch that and set . Scale the vectors again (preserving ) so that the structure equations hold. Let be the simple roots, and set . This fixes the vectors  in the Chevalley basis $\mathrm{ℬ}$. Write  . We need to make one final scaling of the vectors ${X}_{{\mathrm{α}}_{}}$,for. We calculate the structure constants , for and and generate the system of quadratic equations  . Here is the largest positive integer such that is not a root. Put for and solve for the remaining . Finally set  and put   and  for . This completes the construction of the Chevalley basis $\mathrm{ℬ}$' . We have for all , where the matrix ${a}_{\mathrm{ij}}$ is the Cartan matrix for $\mathrm{𝔤}$ and, also,  where . Note that in the Chevalley basis all the structure constants are integers and that the transformation ,  is a Lie algebra automorphism.   See N. Bourbaki, Lie Groups and Lie Algebras, Chapters 7-9, Section 4 for additional details.