WASHINGTON — With a few exceptions in the past half-century, there has been a simple rule of thumb when it comes to international conflict: America does not use force without Britain at its side.

So when Prime Minister David Cameron was unable to muster the votes in Parliament for support for a strike in Syria — even one limited to stopping the future use of chemical weapons — shock could be heard in the voices of senior White House officials who never saw the British rejection coming.

“Bungled by Cameron,” said one.

“Embarrassing,” said another. “For Cameron, and for us.”

Now Mr. Obama is left to cope with miscalculations on both sides of the Atlantic. If he goes ahead with the strike — which seems all but inevitable, based on the statements of senior administration officials who say the president is determined to restore “international norms” against the use of chemical weapons — he will look more isolated than any president in recent memory entering a conflict.

True, Britain stayed out of Vietnam — it was dealing with issues in the Malay Peninsula and Indonesia at the time — and there was no need for Britain in small actions in Panama and Grenada. Ronald Reagan angered his close partner Margaret Thatcher by providing minimal assistance in the Falklands War. But the Middle East, site of Britain’s former empire, is a different matter — territory in which Britain and the United States have long history and deep interests.