Nilpotent Lie algebras of class $4$ with the derived subalgebra of dimension $3$

TL;DR

This paper provides a complete classification of finite-dimensional nilpotent Lie algebras of class 4 with a 3-dimensional derived subalgebra, including identifying which are capable.

Contribution

It offers a full classification of nilpotent Lie algebras of class 4 with derived subalgebra dimension 3, and determines their capability status.

Findings

Complete classification of such Lie algebras

Identification of capable Lie algebras within this class

Structural properties of classified Lie algebras

Abstract

The paper is devoted to give a full classification of all finite dimensional nilpotent Lie algebras $ L $ of class $4$ such that $ \dim L^2=3. $ Moreover, we classify the capable ones.