Prime power order implies nilpotent for Moufang loops

Statement
Suppose $$L$$ is a Moufang loop of prime power order, i.e., a Moufang loop whose underlying set has order a prime power. Then, $$L$$ is a nilpotent Moufang loop (i.e., a finite nilpotent Moufang loop).

Similar facts

 * Prime power order implies nilpotent (in groups)

Opposite facts

 * Prime power order not implies nilpotent for Lie rings