In this new pivot, the Saudis decided that Iranian expansionism has turned the kingdom’s regional fight with the Brotherhood into an unaffordable distraction—especially now that the Brotherhood appears too feeble to imperil the monarchy and its allies. Contributing to this shift is Riyadh’s existential fear of Iranian-fostered unrest within the kingdom’s own Shiite minority.

“The Yemen issue is very central for the Saudis. They fear that if there is a government in line with Iranian foreign policy there, it may give a strong voice for the Shiites inside Saudi Arabia,” said Khalid Almezaini, coordinator of the Gulf studies program at Qatar University.

“The Houthis have military capabilities, weapons, and represent a serious threat. The Muslim Brotherhood is much weaker, and the Saudis now are in a much better position to control the brothers wherever they are,” he added…

By moving from the region’s anti-Brotherhood vanguard closer to the middle ground, Saudi Arabia has managed to rally all the Sunni countries that matter around its Yemen campaign—from Turkey, Mr. Morsi’s key backer, and maverick Qatar to Mr. Sisi, now Egypt’s president, and the staunchly anti-Islamist U.A.E.