Everyone knows that the Conservative and Labour parties are divided. The Tories are split between Cameroons and Traditionalists. Last week, Graham Brady, the chairman of the backbench 1922 Committee that David Cameron tried and failed to abolish, accused the leadership of losing the election by betraying the party's principles.
Well, he didn't put it quite like that, but we got his drift. He said that the 1922 Committee would carry out an inquiry into the Conservative election campaign. We know what that means, because so many backbench Tory MPs subscribe to the latest version of the treachery thesis. This holds that the party failed to win a majority because (a) the Big Society was a pile of pap; (b) the party failed to campaign strongly on the issues on which it was strong, namely immigration, crime and welfare reform; (c) Cameron had dropped grammar schools and the referendum on the Lisbon Treaty; and (d) then he gave Nick Clegg the gift of equal status in the televised debates.
Labour is equally deeply divided, although since the election the party has been surprisingly united in attacking the coalition Government as the reincarnation of Margaret Thatcher in the deceptively winning guise of a civil partnership between Cameron and Clegg. The Blairites don't agree with this characterisation of the enemy, but recognise that David Miliband has to join in to win the votes of Labour's own Traditionalists, while Ed Balls has showed a touch of genius for the politics of opposition that he never showed for the politics of government.
These divisions in the two larger parties are well known. Yet, since the coalition was formed, the tensions in the Liberal Democrats have become important. The cleavage in Nick Clegg's party, while less well known, is deep and bitter. As Henry Kissinger said of student politics, it is so vicious because the stakes are so small. Now, for the Lib Dems, the stakes are higher. Allow me to provide a brief guide.
To simplify, the party is divided between social and economic liberals. To simplify further, between "left" and "right". The social liberals, the "left", are "dominant but intellectually dormant", as James Crabtree puts it in this month's Prospect. Clegg is an economic liberal, identified with The Orange Book, a widely unread collection published in 2004, which became a totem of internal positioning. The untold story of Clegg's leadership, since he narrowly beat the slightly more "left"-wing Chris Huhne, who is actually also an economic liberal, in 2007, is one of repeated and unsuccessful attempts to challenge the party's social liberal wing.
How could you have forgotten Clegg's advocacy of a "new liberal model of schools that are non-selective, under local government strategic oversight but not run by the council", which prefigured Michael Gove's free schools? Or his policy document "Make It Happen", which called for big cuts in public spending and tax cuts for the low paid just before the credit crisis in 2008? Or his attempt to end his party's opposition to tuition fees, which at the end of 2008 finally provoked the social liberals to fight back and organise a slate of candidates to take over the party's federal policy committee?
After the election, both wings of the party united behind the coalition. The dream of a share of power in a hung parliament has kept the party going for all the decades since the Lib-Lab pact of 1977-78. The social liberals were less keen, but the division in the party is only between those who have their doubts and those who are enthusiasts for the coalition.
Now, however, the fun begins. Most of the government jobs are in the hands of economic liberals, even if the most astringent of them, David Laws, didn't last long, while the party machinery is in the hands of the "left". Backbench Lib Dem MPs are mostly caught in the middle, heart with the left, head with the right. Most of them know that the history of their party is one of schism. That every time it has been in a peacetime coalition government, it has split, with the leadership being absorbed into the Conservative Party and the surviving rump condemned to weakness and decline.
Will that happen this time? Absorption is Cameron's objective. He is so keen to stress how well he gets on with his coalition partners that, in private, he confides that Ken "Prison Doesn't Work" Clarke, the Justice Secretary, causes him more difficulties than any of the Liberal Democrats in Cabinet, by being more "radical" than any of them. Clegg's aim, meanwhile, is to avoid being absorbed. But that means that he needs to have somewhere else to go.
The most important point made by Peter Mandelson in one of his interviews yesterday to promote his memoir was to urge the Labour Party not to close the door to a future deal with the Lib Dems, who are not beyond the "progressive pale".
It would be all too easy for Labour leadership candidates to please their natural supporters by accusing the Lib Dems of being collaborators – a word John Prescott used to describe his former colleague John Hutton, advising the Government on pensions – as if we were now living under Nazi occupation. But that risks driving the Lib Dems further into the arms of the Tories. Labour's priority ought to be to ensure that it would be a more attractive coalition partner than the Tory party if there is another hung parliament after the 2015 election.
That means understanding the Liberal Democrats, not least in the sense of understanding their internal tensions and exploiting them. That means, in turn, accepting Labour's responsibility for having lost control of the nation's finances, as a basis for making the Lib Dems uncomfortable, by accusing the Tories not of cutting public spending but of cutting too deeply. Once again, it turns out that David Miliband is the candidate most likely to lead Labour to where it needs to be in five years' time.
John Rentoul blogs at: www.independent.co.uk/jrentoul