Derivation subalgebras of Lie algebras

Let L be a Lie algebra and I, J be two ideals of L. If Der(J)(I) (L) denotes the set of all derivations of L whose images are in I and send J to zero, then we give necessary and sufficient conditions under which Der(J)(I) (L) is equal to some special subalgebras of the derivation algebra of L. We also consider finite dimensional Lie algebra for which the center of the set of inner derivations, Z(IDer(L)), is equal to the set of central derivations of L, Der(z)(L), and give a characterisation of such Lie algebras.