Maintaining Matroid 3-Connectivity With Respect to a Fixed Basis

We show that for any 3-connected matroid M on a ground set of at least four elements such that M does not contain any 4-element fans, and any basis B of M, there exists a set K [is a subset of] E(M) of four distinct elements such that for all k [is an element of the set] K, si(M=k) is 3-connected whenever k [is an element of the set] B, and co(M\k) is 3-connected whenever k [is an element of the set] E(M) - B. Moreover, we show that if no other elements of E(M) - K satisfy this property, then M necessarily has path-width 3.