Image of Lie polynomial of degree $2$ evaluated on Nilpotent Lie algebra

TL;DR

This paper characterizes the image of degree 2 multilinear Lie polynomials evaluated on finite-dimensional nilpotent Lie algebras with small derived algebra dimension.

Contribution

It provides a detailed description of the polynomial images specifically for nilpotent Lie algebras with derived algebra dimension at most 4.

Findings

Explicit description of the polynomial image for the specified class of Lie algebras

Extension of previous results to a broader class of nilpotent Lie algebras

Potential applications in understanding polynomial identities in Lie algebras

Abstract

We delineate the image of multilinear Lie polynomial of degree $2$ evaluated on $L$ where $L$ is a finite-dimensional nilpotent Lie algebra over field $k$ with $\dim L' \leq 4$.