  We continue the study of separable elements in finite Weyl groups. These
elements generalize the well-studied class of separable permutations. We show
that the multiplication map $W/U \times U \to W$ is a length-additive
bijection, or splitting, of the Weyl group $W$ when $U$ is an order ideal in
right weak order generated by a separable element; this generalizes a result
for the symmetric group, answering an open problem of Wei.
  For a generalized quotient of the symmetric group, we show that this
multiplication map is a bijection if and only if $U$ is an order ideal in right
weak order generated by a separable element, thereby classifying those
generalized quotients which induce splittings of the symmetric group, resolving
a problem of Bj\"{o}rner and Wachs from 1988. We also prove that this map is
always surjective when $U$ is an order ideal in right weak order. Interpreting
these sets of permutations as linear extensions of 2-dimensional posets gives
the first direct combinatorial proof of an inequality due originally to
Sidorenko in 1991, answering an open problem Morales, Pak, and Panova. We also
prove a new $q$-analog of Sidorenko's formula. All of these results are
conjectured to extend to arbitrary finite Weyl groups.
  Finally, we show that separable elements in $W$ are in bijection with the
faces of all dimensions of several copies of the graph associahedron of the
Dynkin diagram of $W$. This correspondence associates to each separable element
$w$ a certain nested set; we give product formulas for the rank generating
functions of the principal upper and lower order ideals generated by $w$ in
terms of these nested sets, generalizing several known formulas.
