Jordan algebras at Jordan elements of semiprime rings with involution

In this paper we determine the Jordan algebras associated to ad-nilpotent elements of index at most 3 in Lie algebras R−R− and Skew(R,⁎)Skew(R,⁎) for semiprime rings R without or with involution ⁎. To do so we first characterize these ad-nilpotent elements.