Moufang identities
$$x(y⋅xz)=(xy⋅x)z,$$
$$(zx⋅y)x=z(x⋅yx),$$
$$xy⋅zx=x(yz⋅x)$$
are identities deeply related with alternativity (since setting $z=1$ one recovers left and right alternativity), while a Moufang plane is a strictly geometrical concept that encodes the fact that the group of automorphisms that fixes every point of a line acts transitively on the points of the plane, not on the line.
I know it is a well established fact that Moufang identities are deeply related to the geometric properties of the Moufang plane, but I could not find any direct proof. Can someone explain the main lines of the proof or at least give me a reference that directly addresses the question?



Moufang identities and Moufang plane
This is discussed and proved in detail in *Hall, Marshall jun.*, The theory of groups, New York: The Macmillan Company. xiii, 434 p. (1959). [ZBL0084.02202](https://zbmath.org/?q=an:0084.02202), specifically in chapter 20 "Group Theory and Projective Planes", and in there in section 20.5 "Moufang and Desarguesian Planes".


See also <https://www.ams.org/notices/200710/tx071001294p.pdf> for a quick overview of the idea, esp. the definition of "ternary rings" and how to use them to coordinatize projective planes.


