Towards a splitter theorem for internally 4-connected binary matroids IV

In our quest to find a splitter theorem for internally 4-connected binary matroids, we proved in the preceding paper in this series that, except when M or its dual is a cubic Möbius or planar ladder or a certain coextension thereof, an internally 4-connected binary matroid M with an internally 4-connected proper minor N   either has a proper internally 4-connected minor M′M′ with an N  -minor such that |E(M)−E(M′)|⩽3|E(M)−E(M′)|⩽3 or has, up to duality, a triangle T and an element e of T   such that M\eM\e has an N  -minor and has the property that one side of every 3-separation is a fan with at most four elements. This paper proves that, when we cannot find such a proper internally 4-connected minor M′M′ of M, we can incorporate the triangle T into one of two substructures of M: a bowtie or an augmented 4-wheel. In the first of these, M   has a triangle T′T′ disjoint from T   and a 4-cocircuit D⁎D⁎ that contains e   and meets T′T′. In the second, T is one of the triangles in a 4-wheel restriction of M with helpful additional structure.