Moufang loop

In terms of Moufang's identities
A Moufang loop is a loop $$L$$ with multiplication $$*$$ satisfying the following three identities:


 * 1) $$\! z * (x * (z * y)) = ((z * x) * z) * y \ \forall \ x,y,z \in L$$
 * 2) $$\! x * (z * (y * z)) = ((x * z) * y) * z \ \forall \ x,y,z \in L$$
 * 3) $$\! (z * x) * (y * z) = (z * (x * y)) * z \ \forall \ x,y,z \in L$$

In terms of Bol loops
A Moufang loop is a loop that is both a defining ingredient::left Bol loop and a defining ingredient::right Bol loop.