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