  We define ``star reducible'' Coxeter groups to be those Coxeter groups for
which every fully commutative element (in the sense of Stembridge) is
equivalent to a product of commuting generators by a sequence of
length-decreasing star operations (in the sense of Lusztig). We show that the
Kazhdan--Lusztig bases of these groups have a nice projection property to the
Temperley--Lieb type quotient, and furthermore that the images of the basis
elements $C'_w$ (for fully commutative $w$) in the quotient have structure
constants in ${\Bbb Z}^{\geq 0}[v, v^{-1}]$. We also classify the star
reducible Coxeter groups and show that they form nine infinite families (types
$A_n$, $B_n$, $D_n$, $E_n$, $F_n$, $H_n$, affine $A_{n-1}$ for $n$ odd, affine
$C_{n-1}$ for $n$ even, and the case where the Coxeter graph is complete), with
two exceptional cases (of ranks 6 and 7).
  This paper is the sequel to math.QA/0509362.
