Chevalley basis

In mathematics, a Chevalley basis for a simple complex Lie algebra is a basis constructed by Claude Chevalley with the property that all structure constants are integers. Chevalley used these bases to construct analogues of Lie groups over finite fields, called Chevalley groups. The Chevalley basis is the Cartan-Weyl basis, but with a different normalization.
The generators of a Lie group are split into the generators "H" and "E" indexed by simple roots and their negatives formula_1. The Cartan-Weyl basis may be written as
Defining the dual root or coroot of formula_4 as
One may perform a change of basis to define
The Cartan integers are
The resulting relations among the generators are the following:
where in the last relation formula_12 is the greatest positive integer such that formula_13 is a root and we consider formula_14 if formula_15 is not a root.
For determining the sign in the last relation one fixes an ordering of roots which respects addition, i.e., if formula_16 then formula_17 provided that all four are roots. We then call formula_18 an extraspecial pair of roots if they are both positive and formula_19 is minimal among all formula_20 that occur in pairs of positive roots formula_21 satisfying formula_22. The sign in the last relation can be chosen arbitrarily whenever formula_18 is an extraspecial pair of roots. This then determines the signs for all remaining pairs of roots.