  Following Lusztig, we consider a Coxeter group $W$ together with a weight
function. Geck showed that the Kazhdan-Lusztig cells of $W$ are compatible with
parabolic subgroups. In this paper, we generalize this argument to some subsets
of $W$ which may not be parabolic subgroups. We obtain two applications: we
show that under specific technical conditions on the parameters, the cells of a
certain finite parabolic subgroup of $W$ are cells in the whole group, and we
decompose the affine Weyl group $\tilde{G}_{2}$ into left and two-sided cells
for a whole class of weight functions.
