For a matroid M, an element e such that both M\e and M/e
are regular is called a regular element of M. We determine completely the
structure of non-regular matroids with at least two regular elements. Besides
four small size matroids, all 3-connected matroids in the class can be pieced
together from F7​ or S8​ and a regular matroid using 3-sums. This result
takes a step toward solving a problem posed by Paul Seymour: Find all
3-connected non-regular matroids with at least one regular element [5, 14.8.8]