Finite-Dimensional Nilpotent Lie Algebras with Central Derivation Lie Algebras of Minimal Possible

Let L be a finite-dimensional nilpotent Lie algebra and let Der(z)(L) be the set of all central derivations of L. It is a well-known fact that Z(Inn(L)), the center of the inner derivation of a Lie algebra L, is contained in Der(z)(L). We investigate finite-dimensional nilpotent Lie algebras, say L, that Der(z)(L) = Z(Inn(L)). Moreover, we characterize these Lie algebras with some conditions, such as C-L (Z(L-2)) not equal L-2, L having a maximal abelian subalgebra, Camina Lie algebras, and Lie algebras with coclass at most four.