Decomposition of 3-connected representable matroids

In this paper, we prove that via an operation “reducing”, every 3-connected representable matroid M with at least nine elements can be decomposed into a set of sequentially 4-connected matroids and three special matroids which we call freely-placed-line matroids, spike-like matroids and swirl-like matroids; more concretely, there is a labeled tree that gives a precise description of the way that M built from its pieces.