People say grammar schools are divisive, but they’re wrong: they are the only policy on which you can unite Jeremy Corbyn, Michael Gove, the Liberal Democrats, Policy Exchange, the head of Ofsted and the teaching unions.

The consensus among policy thinkers is near-universal: selection at 11 doesn’t work. Even in Germany, in many ways the model to which a raft of politicians – most notably Ed Miliband – aspire to emulate, their selective model is one of the biggest drags on social mobility.

On left and right, though there is a fierce argument about how you get there, there is a consensus among policymakers that the ingredients of a good school are clear: high-quality teachers and effective leadership.

And we forget now, when grammar schools consistently attract the support of a plurality and in some polls a majority of voters, that by the time that Anthony Crosland triggered their abolition, they had become toxic with most voters, because the losers from the grammar school system outnumbered the winners.

So the policy argument in favour of a return to grammar schools is stacked heavily against them, making it a tricky fight for Downing Street.

But park the policy for a moment – what about the politics? Passing new grammar schools through the Houses of Parliament – which, thanks to Labour’s decision in 1998 to ban the creation of new grammars, would be necessary – is a tricky task.

The government doesn’t have a manifesto commitment to reintroduce grammar schools which means that the House of Lords can vote it down if it so chooses – the government doesn’t have a majority in the upper house and it’s difficult to see where it would find one.

The government could pass it by use of the Parliament Act, which means the Lords can only delay a bill by a year – last used to ban fox hunting back in 2004 – but that requires passing it through the House of Commons, which may be trickier than it looks.

The government only has a majority of 12, something which both Downing Street and Whitehall generally are all too aware. Passing policy with such a small majority is unfamiliar to all but a small handful of very old hands – for the last 19 years every incumbent government has had a large majority, from Labour’s landslide years to the 77-seat majority enjoyed by the coalition government – and that, as well as May’s dislike of insubstantial legislation in search of a quick headline, is one reason why what comes before Parliament will be the subject of great care.

The re-creation of grammar schools is a significant repudiaton for the Gove agenda that every school should achieve excellence, and if the former Secretary of State lends his voice, that will embolden MPs on the Conservative left to vote against the measure. Gove’s incentive to do so would be personal as well as political. There is no love lost between he and May, who as well as disagreeing on substantial matters of policy have wildly divergent approaches to politics.

But – and this is one of the wider difficulties for the Osborne-Gove tendency, purged en masse when May put her government together – although being defeated over grammar schools won’t have anyone opening the champagne in Number 10, being the architect of that loss would near-certainly bring about the end of Gove’s slim hopes of a return to the top table of politics.

If there is a path back to relevancy for Gove and his allies, it almost certainly lies with the Tory right, who, like Gove in his appearances in the chamber thus far, support a swifter – and more economically damaging – version of Brexit than that favoured by Theresa May. But those would-be allies are also die-hard supporters of a return to grammars.

So what Gove does over the coming days and weeks will be instructive – does he attempt to defend his existing political legacy, or abandon it in search of the longshot return to the top table?