On minors of non-binary matroids

No binary matroid has a minor isomorphic to U ₄ ², the "four-point line", and Tutte showed that, conversely, every non-binary matroid has a U ₄ ² minor. However, more can be said about the element sets of U ₄ ² minors and their distribution. Bixby characterized those elements which are in U ₄ ² minors; a matroid M has a U ₄ ² minor using element x if and only if the connected component of M containing x is non-binary. We give a similar (but more complicated) characterization for pairs of elements. In particular, we prove that for every two elements of a 3-connected non-binary matroid, there is a U ₄ ² minor using them both.