  This paper is part of the program to classify Kazhdan-Lusztig cells for Weyl
groups of type $D_n$. We prove analogous results to those of section 4 of
Kazhdan-Lusztig's original paper, this time related to a parabolic subgroup of
type $D_4$. We also show how this is used in the definition of the generalized
$\tau$-invariant.
