The Americans knew a lot was riding on a Sept. 11 meeting with the king of Saudi Arabia at his summer palace on the Red Sea.

A year earlier, King Abdullah had fumed when President Barack Obama called off strikes against the regime of Syria’s Bashar al-Assad. This time, the U.S. needed the king’s commitment to support a different Syrian mission—against the extremist group Islamic State—knowing there was little hope of assembling an Arab front without it.

At the palace, Secretary of State John Kerry requested assistance up to and including air strikes, according to U.S. and Gulf officials. “We will provide any support you need,” the king said. […]