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.　Crown　Copyright　(C)　2011　Published　by　Elsevier　Inc.　All　rights　reserved.