John Stembridge has observed that at least one example of an irreducible component from each of the 15 classes may be produced with the construction described below. The efficiency of this observation is remarkable - his construction must be executed only 19 times to see at least one example of each of the 15 classes occuring "in nature". Although this comment per se is not in his paper, "Quasi-minuscule quotients and reduced words for reflections", the general version of the following construction does play a prominent role in that paper.
Let beta be the highest root of a finite root system of rank q. As i runs from 1 to q, let w(i) be the shortest element of the Weyl group of the root system such that w(i).beta = alpha(i), the ith positive simple root. Then w(i) is lambda-minuscule, with lambda = beta. Let P(i) be the set of positive co-roots which are taken to negative co-roots by w(i). Place the dual of the standard order by simple co-roots on P(i). Using results of Stembridge's "Minuscule elements of Weyl groups" and "On the fully commutative elements of Coxeter groups" and our "Minuscule" paper, it can be shown that P(i) will be d-complete. For the root systems E(q) as q ranges from 6 to 8, the top tree of each P(i) is the Dynkin diagram E(q). Here the top tree has q-4 nodes in its neck, 1 node in its left branch, and 2 nodes in its right branch. The number of distinct P(i) which arise are respectively 4, 7, and 8 for E(6), E(7), and E(8). Of these 19 posets, 14 are maximal irreducible components. Twelve classes (2 - 7 and 9 - 14) are represented once apiece by these 14 posets; Class 8 is represented twice. Another 3 of the 19 posets are slant sums of two slant irreducible posets Q and P. Here Q is the one element poset and P is a maximal irreducible component from Classes 1, 2, and 15 respectively.