Matroids with at least two regular elements

For a matroid MM, an element ee such that both M∖eM∖e and M/eM/e are regular is called a regular element of MM. 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 F7F7 or S8S8 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 Oxley (1992) [5, 14.8.8].