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 , where , is a basis for a Cartan subalgebra, and where , (the positive roots), gives a root space decomposition for . By definition of a split, semi-simple Lie algebra, the root vectors are all real. Let 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 . Write
.
We need to make one final scaling of the vectors ,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 ' . We have
for all , where the matrix is the Cartan matrix for 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.
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