The computation of structure constants according to Jacques Tits

Suppose g to be a complex semisimple Lie algebra. In 1955, Chevalley showed that one can assign to it an essentially unique Zstructure, with structure constants of a simple form, themselves unique up to some choices of sign. An algorithm to compute these is also due implicitly to Chevalley, explained explicitly in [Carter:1972] and more recently in [CohenMurrayTaylor:2004]. It is reasonably efficient, and has been implemented in the computer algebra programMAGMA. The principle behind the algorithm is simple—it is based on a straightforward transcription of the Jacobi identity. [CohenMurrayTaylor:2004] goes on to explain in detail how to compute in semisimple algebraic groups associated to the Lie algebra in terms of the Bruhat decomposition. Another account of structure constants was presented in [FrenkelKac:1980] for simply laced systems, and this has been extended to all cases by [Rylands:2000], using the technique of defining the other systems by ‘folding ’ ones that are simply laced. These algorithms are all very efficient, and work ell in practice. But there are reasons for trying another approach to the computation of structure constants. The most obvious is that, as far as I can tell, they do not extend to arbitrary KacMoody algebras. Although there are obvious problems due to the infinite structure of the root system, the really essential reason is the interference caused by imaginary roots.