Moufang implies alternative

Statement
Any Moufang loop is an alternative loop.

Stronger facts

 * Moufang implies diassociative

Applications

 * Moufang implies diassociative

Moufang loop
A Moufang loop is a loop $$L$$ satisfying the following identities for all $$x,y,z \in L$$ (where two or more of the $$x,y,z$$ could possibly be equal):


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

Alternative loop
An alternative loop is a loop $$L$$ satisfying the following two identities for all $$a,b \in L$$ (where $$a,b$$ may be equal or distinct):


 * $$a * (a * b) = (a * a) * b$$
 * $$a * (b * b) = (a * b) * b$$

Proof
Given: A Moufang loop $$L$$ with identity element $$e$$.

To prove: For all $$a,b \in L$$ (possibly equal), we have $$a * (a * b) = (a * a) * b$$ (left alternative law) and $$a * (b * b) = (a * b) * b$$ (right alternative law).

Proof: For the left alternative law, set $$x = e$$, $$y = b$$, and $$z = a$$ in Moufang's identity (1) given above.

For the right alternative law, set $$x = a$$, $$y = e$$, and $$z = b$$ in Moufang's identity (2) given above.