A note on commutators in compact semisimple Lie algebras

TL;DR

This paper proves that in compact semisimple Lie algebras, any two elements can be expressed as commutators with a common regular element, and shows that subspaces of codimension at most 2 contain a Cartan subalgebra.

Contribution

It establishes a new relationship between elements and regular elements in compact semisimple Lie algebras, and demonstrates the existence of Cartan subalgebras in low-codimension subspaces.

Findings

Any two elements are commutators with a common regular element.

Subspaces of codimension ≤ 2 contain a Cartan subalgebra.

Provides a new structural insight into compact semisimple Lie algebras.

Abstract

Given two elements $A,B$ in a compact semisimple Lie algebra, we show that there is a regular element $X$ and elements $Y,Z$ with $A=[X,Y]$ and $B=[X,Z]$. In the course of the proof we show also that every linear subspace $V$ of codimension at most 2 in the Lie algebra contains a CSA.