  This paper is a report on a computer check of some positivity properties of
the Hecke algebra in type H4, including the non-negativity of the coefficients
of the structure constants in the Kazhdan-Lusztig basis. This answers a
long-standing question of Lusztig's. The same algorithm, carried out by hand,
also allows us to deal with the case of dihedral Coxeter groups.
