The system satisfies the conditions Let K be a field and X the projective line P1(K) over K. Let Ux be the stabiliser of each point x in the group PSL2(K).
The Moufang set determines K up to isomorphism or anti-isomorphism: an application of Hua's identity.
A quadratic Jordan division algebra gives rise to a Moufang set structure.
If U is the quadratic map on the unital algebra J, let τ denote the permutation of the additive group (J,+) defined by Then τ defines a Moufang set structure on J.
The Hua maps ha of the Moufang structure are just the quadratic Ua (De Medts & Weiss 2006).