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

Let M be a 3-connected binary matroid; M is internally 4connected if one side of every 3-separation is a triangle or a triad, and M is (4, 4, S)-connected if one side of every 3-separation is a triangle, a triad, or a 4-element fan. Assume M is internally 4-connected and that neither M nor its dual is a cubic Möbius or planar ladder or a certain coextension thereof. Let N be an internally 4-connected proper minor of M . Our aim is to show that M has a proper internally 4-connected minor with an N -minor that can be obtained from M either by removing at most four elements, or by removing elements in an easily described way from a special substructure of M . When this aim cannot be met, the earlier papers in this series showed that, up to duality, M has a good bowtie, that is, a pair, {x1, x2, x3} and {x4, x5, x6}, of disjoint triangles and a cocircuit, {x2, x3, x4, x5}, where M\x3 has an N minor and is (4, 4, S)-connected. We also showed that, when M has a good bowtie, either M\x3, x6 has an N -minor and M\x6 is (4, 4, S)-connected; or M\x3/x2 has an N -minor and is (4, 4, S)-connected. In this paper, we show that, when M\x3, x6 has no N -minor, M has an internally 4-connected proper minor with an N -minor that can be obtained from M by removing at most three elements, or by removing elements in a well-described way from a special substructure of M . This is the penultimate step towards obtaining a splitter theorem for the class of internally 4-connected binary matroids.