A characterisation of Lie algebras using ideals and subalgebras

TL;DR

This paper characterizes Lie algebras within non-associative algebras by showing that certain ideal and subalgebra properties uniquely identify the variety of all Lie algebras.

Contribution

It establishes a characterization of Lie algebras based on properties of subalgebras and ideals in free algebras, providing a new criterion for identifying Lie algebras.

Findings

Subalgebras of free algebras are free in the characterized variety.

The square of an ideal is an ideal in this variety.

The characterized variety coincides with all Lie algebras.

Abstract

We prove that if, for a nontrivial variety of non-associative algebras, every subalgebra of every free algebra is free and $I^2$ is an ideal whenever $I$ is an ideal, then this variety coincides with the variety of all Lie algebras.