Among the followers of this creed, the battle between Mr. Zawahiri and Mr. Baghdadi is a pivotal development. “Like the old saying goes: The revolution devours its own,” says Jérôme Drevon, a fellow at the Swiss National Science Foundation who specializes in Islamist movements. “What we are seeing is a generational split between older jihadis who have learned pragmatic lessons [of overreach]…and a younger, more brutal generation who don’t believe in or haven’t lived long enough to learn those lessons.”…

The dispute centers on Mr. Zawahiri’s belief that a caliphate—a state that can demand allegiance from all Muslims and declare jihad against the enemies of the faith—can emerge only after the wider Muslim world has been purified. Mr. Zawahiri hopes to bring Muslims out of their unredeemed state of jahiliyya—the type of spiritual ignorance that existed before the Prophet—by excising all contact with corrupting Western influences and placing governing institutions in the hands of administrators who share this vision and can promulgate it to the mass of Muslims…

But Mr. Baghdadi and his followers reject this doctrine of an evolving religious and social consensus. They believe instead that a pure Islamic regime can be more swiftly imposed by force. This basic split has existed for a decade between al Qaeda and its one-time offshoot in Iraq, which formed after the U.S. invaded in 2003 and helped establish the first Shiite government in Iraq in centuries.