Identities in Moufang sets

Moufang sets were introduced by Jacques Tits as an axiomatization of the buildings of rank one that arise from simple algebraic groups of relative rank one. These fascinating objects have a simple definition and yet their structure is rich, while it is rigid enough to allow for (at least partial) classification. In this paper we obtain two identities that hold in any Moufang set. These identities are closely related to the axioms that define a quadratic Jordan algebra. We apply them in the case when the Moufang set is so-called special and has abelian root groups. In addition we push forward the theory of special Moufang sets.