  We investigate the relation between the structure of a Moufang loop and its
inner mapping group. Moufang loops of odd order with commuting inner mappings
have nilpotency class at most two. $6$-divisible Moufang loops with commuting
inner mappings have nilpotency class at most two. There is a Moufang loop of
order $2^{14}$ with commuting inner mappings and of nilpotency class three.
