Al Qaeda’s leadership of Islamist militancy in Yemen is being challenged amid the chaos of civil war by its rival Islamic State, which could become a bigger threat to the group than the U.S. drones that periodically kill its commanders.

Al Qaeda in the Arabian Peninsula (AQAP) remains the country’s top militant network and its several thousand fighters and bomb makers, who have repeatedly tried to bring down Western aircraft, are still seen as a serious threat by Washington.

But followers of Islamic State in Yemen have sought to steal al Qaeda’s thunder by launching a string of attacks against the Zaydi Shi’ite Muslim Houthi group, which is fighting a messy war against a Saudi-led coalition of Arab counties. […]