After 64 years of rule by sons of King Ibn Saud, Saudi Arabia is making a transition to a new generation of leaders. The ailing 81-year-old King Salman decreed in June that his successor would be his 31-year-old son, Mohammed bin Salman, who is already largely running the country.

The challenge of succession has been hanging over Saudi rulers for decades—ensuring that the family’s amazing unity continued beyond the generation of King…