# Chevalley basis

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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 generators of a Lie group are split into the generators H and E such that:

$[H_{\alpha_i},H_{\alpha_j}]=0$
$[H_{\alpha_i},E_{\alpha_j}]=A_{ij}E_{\alpha_j}$
$[E_{\alpha_i},E_{\alpha_j}]=H_{\alpha_j}$
$[E_{\beta},E_{\gamma}]=\pm(p+1)E_{\beta+\gamma}$

where p = m if β + γ is a root and m is the greatest positive integer such that γ − mβ is a root.