So much scandal is swirling around President Obama that it was hard to spot what must be the biggest strategic error of the week — his warning to Prime Minister Cameron that if Britain leaves the European Union it could lose clout in Washington. The story ran in few, if any, places other than the London Financial Times, which featured on page one a picture of Messrs. Cameron and Obama in the Oval Office. The headline read, “Obama warns Cameron that Britain would lose influence in the US if it pulls out of EU.”

Who in the world came up with that brainstorm? The idea seems to be, as the FT quotes Mr. Obama as articulating it for Mr. Cameron, that Britain’s membership in the EU is “an expression of its influence and its role in the world.” The president advised Mr. Cameron, in public statements yesterday, to try to “fix what’s broken” in the European Union rather than pull out. In context that’s an intervention by Mr. Obama into Britain’s domestic political situation, where a fast-growing political party is challenging Mr. Cameron’s government over the issue of Europe.

The challenger is the United Kingdom Independence Party. It was founded in 1993 in the wake of the Maastricht Treaty, which led to the creation of the European Union and the Euro. For years a marginal group, with a libertarian streak, UKIP has been surging lately. It has helped box Mr. Cameron in to promising a referendum on whether Britain should stay in the European Union, and if recent polls are an indication, there’s a fair chance that Britons would decide to exit the socialist satrapy that has been set up in the years since Maastricht.

So when did it become American policy to set itself against the British voters? A British exit — known as “Brexit” in British shorthand — ought to be seen as an opportunity for America. These columns have been making this point for some years, urging the idea that an exit of Britain from the EU would present a chance to forge something substantive out of the “special relationship” that Britain and America are supposed to enjoy. It’s a situation that calls for creative thinking in the White House and the State Department.

Yet somehow the logic of building a more substantial alliance between America and Britain has eluded Mr. Obama, who, along with Secretary Clinton, has run a foreign policy devoid of new ideas, save for a marked coolness toward Britain. He made this obvious from the start of his presidency, when he removed from the Oval Office the bust of Winston Churchill, who, for the record, was made an American citizen, if an honorary one, by an act of Congress and who brought the special relationship to its apogee.

It is hard to imagine that Churchill would be anything but horrified at the slackness of relations between Britain and America today. Little more than a year into Obama’s presidency things were so bad that a group of parliamentarians at London actually called on the British foreign office to stop using the phrase “special relationship.” One former British envoy in Washington, David Manning, was quoted as saying of Mr. Obama: “The sentimental reflexes, if you like, are not there.”

By our lights what has made our relationship with Britain special has been neither blood ties nor sentiment but a shared concept of individual liberty. That is, ideas going back to people like John Locke and Adam Smith. These are the ideas under threat in the European Union. They are the ideas Prime Minister Margaret Thatcher was defending when, in one of her most famous speeches, she said, at Bruges, Belgium, that her government had not “rolled back the frontiers of the state in Britain, only to see them re-imposed at a European level with a European super-state exercising a new dominance from Brussels.”

There are those who reckon that a British American pact of some sort might be a good idea for America but who wonder what is in it for Britain, given the weakness of the American dollar, the enormous American debt, and the ideas of the Obama administration. Throw in Obamacare and the regulatory state and a penchant for military disengagement and it might be better — the argument goes — for Britain to look to, say, Canada, which has had a particularly successful conservative government under Prime Minister Harper.

There are those of us, however, who nurse hopes that Democratic control of the White House and the Senate won’t last forever. The idea of a pro-growth, sound-money, pro-democracy grouping comprising the countries that most cherish the ideas of liberty, limited government, and free markets — well, it’s something. It’s a basis for a foreign policy. It’s a sharp contrast with Europe. It would be better advice for an American president to give to the prime minister of Britain than to suggest he go on dickering with the dirigistes at Brussels.