  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.
