Lie-central derivations, Lie-centroids and Lie-stem Leibniz algebras

In this paper, we introduce the notion of a $\mathsf{Lie}$-derivation. This concept generalizes derivations for non-$\mathsf{Lie}$ Leibniz algebras. We study these $\mathsf{Lie}$-derivations in the case where their image is contained in the $\mathsf{Lie}$-center, and call them $\mathsf{Lie}$-central derivations. We provide a characterization of $\mathsf{Lie}$-stem Leibniz algebras by their $\mathsf{Lie}$-central derivations, and prove several properties of the Lie algebra of $\mathsf{Lie}$-central derivations for $\mathsf{Lie}$-nilpotent Leibniz algebras of class 2. We also introduce ${\sf ID}_*$-$\mathsf{Lie}$-derivations. An ${\sf ID}_*$-$\mathsf{Lie}$-derivation of a Leibniz algebra $\mathfrak{g}$ is a $\mathsf{Lie}$-derivation of $\mathfrak{g}$ in which the image is contained in the second term of the lower $\mathsf{Lie}$-central series of $\mathfrak{g}$, and which vanishes on $\mathsf{Lie}$-central elements. We provide an upper bound for the dimension of the Lie algebra ${\sf ID}_*^{\mathsf{Lie}}(\mathfrak{g})$ of ${\sf ID}_*$-${\mathsf{Lie}}$-derivation of $\mathfrak{g}$, and prove that the sets ${\sf ID}_*^{\mathsf{Lie}}(\mathfrak{g})$ and ${\sf ID}_*^{\mathsf{Lie}}(\mathfrak{q})$ are isomorphic for any two $\mathsf{Lie}$-isoclinic Leibniz algebras $\mathfrak{g}$ and $\mathfrak{q}$.