The Bush administration didn't invade the country for the purpose of occupying it as a colony in the matter of 19th century Britain and France, for whom divide and rule tactics served that end. The US invaded Iraq to install a bourgeois puppet government under Chalabi, which they were confident would rest on the stable support of the grateful Shia majority. The Bushites were confident they would have have the invasion force home well before Christmas - ready, if necessary, for other "missions" elsewhere against the "axis of evil". That was Rumsfeld's answer to the generals who told him the thin US garrison was incapable of occupying the country.
The overconfident Bush administration, of course, made two fatal miscalculations: a) that the Shia masses under their religious leaderships, once liberated from Baathist repression, would passively accept to be ruled by secular bourgeois exiles rather than push forward to sweep them aside and press their own demands b) that the Sunnis would not reorganize and mount a fierce resistance, a resistance encouraged by the US acceptance of Shia demands for "deBaathification" and the disbandment of the regular Iraqi army - a decision belatedly recognized as a major strategic blunder by the US. The US has been trying ever since to get these genies back in the bottle, primarily through efforts to disband the independent militias on both sides. It has fought not only the Sunni insurgents, but also tried to bloodily suppress the Sadrist movement before the Sadrists became too powerful to be suppressed.
The invasion unleashed separatist tendencies which were latent in the country, but these have developed in accordance with their own logic rather than the encouragement of the occupying power. The core of US strategy over the past three years has been to reintegrate the displaced Sunnis into a tripartite power-sharing arrangement with the triumphant Shias and Kurds rather than further separate the three communities.
The Biden-Gelb plan, which some have mistaken as a US "divide and rule" strategy is, in fact, a response to these fissiparous pressures - an attempt to salvage a unified Iraq by granting the ethnic-based regions the widest possible degree of autonomy within a federal structure. The difficulty, of course, is finding a formula for power- and revenue-sharing which will satisfy all parties to the point they are prepared to merge their militias into a national army.
It is impossible to know, at this point, whether the Americans and Iranians, either jointly or in parallel, will be able to force unity on the factions or whether the process of disintegration has gone so far that only civil war can now decide the future distribution of power and resources. The latter now seems more likely, which is why there is such panic in Washington.
Anyone who believes the Americans went into Iraq with the intention of splitting the country in three in order to rule each part more effectively has to ask themselves this question: Why they did not simply do this by decree or through the constitution they imposed following their successful invasion - before the rapid growth of competing militias plunged the country into chaos, humbled the US military, and politically destroyed the Bush administration?