Jordan supersystems related to Lie superalgebras

Given a Lie superalgebra and an even ad-nilpotent element of index <= 3, one can obtain a Jordan superalgebra attached to that element; inspired by that construction we build a Jordan superpair attached to an odd ad-nilpotent element of index <= 4: We introduce inner ideals for Lie super-algebras, and we prove that the associated subquotients are Jordan superpairs. All three constructions agree when considering abelian inner ideals generated by one element.