Moufang loops with commuting inner mappings

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.