A remark on a proof of $[G, L]=0$ for a Lie group $G$

TL;DR

This paper refines a previous proof showing that the commutator of a compact framed Lie group and its Lie algebra is zero, by optimizing the choice of a circle subgroup within the group.

Contribution

It introduces an improved method for selecting a circle subgroup to simplify the proof of the commutation relation in compact framed Lie groups.

Findings

Enhanced proof of $[G, L]=0$ for compact framed Lie groups

Better understanding of subgroup selection impacts proof simplicity

Potential for broader applications in Lie group theory

Abstract

In this note we give an improvement of our proof of $[G, L]=0$ for a compact framed Lie group $(G, L)$, which depends heavily on the choice of a circle subgroup $S\subset G$. We attempt here to make a more suitable choice of this circle subgroup.