Regular matroids are binary matroids with no minors isomorphic to the Fano matroid or its dual. The Fano matroid is the binary projective plane PG(2, 2). Seymour proved that 3-connected regular matroids are either graphs, cographs, or a special 10-element matroid R10 called a splitter, or else can be decomposed along a non-minimal exact 3-separation induced by another special matroid R12 called a 3-decomposer. Quasiregular matroids are binary matroids with no minor isomorphic to E4, where E4 is a 10-element rank 5 self-dual binary matroid. The class of quasiregular matroids properly contains the class of regular matroids. In a paper that just appeared in the Electronic Journal of Combinatorics (Vol. 25, Issue 3 (2018)), we decomposed the class of quasiregular matroids in a manner similar to regular matroids by showing that quasiregular matroids are either graphs, cographs, or deletion-minors of PG(3,2), R17 or M12 or else can be decomposed along a non-minimal exact 3-separation induced by R12, P9, or its dual. The matroid M12 is a splitter and PG(3,2) and R17 are the maximal 3-connected rank 4 and 5 members, respectively. The class of quasiregular matroids is one of very few excluded minor classes whose members have been described. After Seymour's decomposition of regular matroids, the Fano dual became the starting point for the analysis of binary non-graphic and non-cographic matroids. As a consequence of the decomposition of quasiregular matroids the starting point for the analysis of binary non-graphic and non-cographic matroids is E4.