Abstract:We contrast the simple proof that a quasigroup which satisfies the Moufang identity $(x\cdot yz)x = xy\cdot zx$ is necessarily a loop (Moufang loop) with the remarkably involved prof that a quasigroup which satisfies the Moufang identity $(xy\cdot z)y=x(y\cdot zy)$ is likewise necessarily a Moufang loop and attempt to explain why the proofs are so different in complexity.
Keywords: Moufang quasigroups, Moufang loops, identities of Bol-Moufang type
AMS Subject Classification: 20N05