Minimal faithful representations of abelian Jordan algebras and Lie superalgebras

Over an algebraically closed field of characteristic zero, all the abelian subalgebras of the maximal dimension are classified for any special Jordan algebra. As a consequence, the minimal dimension of the faithful representations of any finite-dimensional abelian Jordan algebra is determined and the minimal faithful representations are classified for the so-called nice abelian Jordan algebras. The same is done for the purely odd Lie superalgebras.