Regular bi-interpretability and finite axiomatizability of Chevalley groups

In this paper we consider Chevalley groups over commutative rings with [Formula: see text], constructed by irreducible root systems of rank [Formula: see text]. We always suppose that for the systems [Formula: see text] our rings contain [Formula: see text] and for the system [Formula: see text] also [Formula: see text]. Under these assumptions we prove that the central quotients of Chevalley groups are regularly bi-interpretable with the corresponding rings, the class of all central quotients of Chevalley groups of a given type is elementarily definable and even finitely axiomatizable (see Definition ??). The same holds for adjoint Chevalley groups and for bondedly generated Chevalley groups. We also give an example of Chevalley group with infinite center, which is not bi-interpretable with the corresponding ring and is elementarily equivalent to a group that is not a Chevalley group itself.