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A note on commutators in compact semisimple Lie algebras

Linus Kramer

Vol. 20 (2023), No. 2-3, 317–324
Abstract

Given any two elements A,B in a compact semisimple Lie algebra, we show that there exist elements X,Y,Z such that

A = [X,Y ] andB = [X,Z].

The proof uses Cartan subalgebras and their root systems. We also review some related problems about Cartan subalgebras in compact semisimple Lie algebras.

Dedicated to Jacques Tits

Keywords
Lie algebra, commutator, Cartan subalgebra
Mathematical Subject Classification
Primary: 22E60
Milestones
Received: 23 August 2022
Revised: 21 March 2023
Accepted: 14 May 2023
Published: 13 September 2023
Authors
Linus Kramer
Mathematisches Institut
Fachbereich Mathematik und Informatik
Universität Münster
Münster
Germany