Computation in Coxeter groups II. Minimal roots
  In the recent paper (Casselman, 2001) I described how a number of ideas due to Fokko du Cloux and myself could be incorporated into a reasonably efficient program to carry out multiplication in arbitrary Coxeter groups. At the end of that paper I discussed how this algorithm could be used to build the reflection table of minimal roots, which could in turn form the basis of a much more efficient multiplication algorithm. In this paper, following a suggestion of Bob Howlett, I explain how results due to Brigitte Brink can be used to construct the minimal root reflection table directly and more efficiently. 
Bill Casselman