A minor-based characterization of matroid 3-connectivity

It is well known that a matroid is 2-connected if and only if every 2-element set is contained in a circuit, or equivalently, a U1,2-minor. This paper proves that a matroid is 3-connected if and only if every 4-element set is contained in a minor isomorphic to a wheel of rank 3 or 4; a whirl of rank 2, 3, or 4; or the relaxation of a rank-3 whirl. Some variants of this result are also discussed.