On pointwise inner derivations of Lie algebras

Let [Formula: see text] be a Lie algebra and [Formula: see text] and [Formula: see text] be the set of all derivations and inner derivations of [Formula: see text], respectively. A derivation [Formula: see text] of a Lie algebra [Formula: see text] is pointwise inner if [Formula: see text] for all [Formula: see text]. The set of all pointwise inner derivations of Lie algebra [Formula: see text] denoted by [Formula: see text] form a subalgebra of [Formula: see text] containing [Formula: see text]. In this paper, we prove that, if [Formula: see text] is nilpotent of class [Formula: see text] (solvable of length [Formula: see text]), then [Formula: see text] is nilpotent of class [Formula: see text] (solvable of length [Formula: see text] or [Formula: see text]). We also prove that if [Formula: see text] is nilpotent of class [Formula: see text], then [Formula: see text] is nilpotent of class at most [Formula: see text], in which [Formula: see text] and [Formula: see text] is the [Formula: see text]th term in the upper central series of [Formula: see text].