# Almost inner derivations of Lie algebras II

## TL;DR

This paper extends the algebraic understanding of almost inner derivations in Lie algebras, explicitly determining these derivations for various classes including free nilpotent, almost abelian, and certain filiform nilpotent Lie algebras over characteristic zero fields.

## Contribution

It provides explicit descriptions of almost inner derivations for several classes of Lie algebras and introduces a family of characteristically nilpotent filiform Lie algebras with all derivations almost inner.

## Key findings

- Determined almost inner derivations for free nilpotent Lie algebras.
- Identified all derivations as almost inner for a family of filiform nilpotent Lie algebras.
- Compared almost inner derivations over different base fields.

## Abstract

We continue the algebraic study of almost inner derivations of Lie algebras over a field of characteristic zero and determine these derivations for free nilpotent Lie algebras, for almost abelian Lie algebras, for Lie algebras whose solvable radical is abelian and for several classes of filiform nilpotent Lie algebras. We find a family of $n$-dimensional characteristically nilpotent filiform Lie algebras $\mathfrak{f}_n$, for all $n\ge 13$, all of whose derivations are almost inner. Finally we compare the almost inner derivations of Lie algebras considered over two different fields $K\supseteq k$ for a finite-dimensional field extension.