Commutators and Cartan subalgebras in Lie algebras of compact semisimple Lie groups

Abstract

First we give a new proof of Goto's theorem for Lie algebras of compact semisimple Lie groups using Coxeter transformations. Namely, every $x$ in $L = \operatorname{Lie}(G)$ can be written as $x =[a, b]$ for some $a$, $b$ in $L$. By using the same method, we give a new proof of the following theorem (thus avoiding the classification tables of fundamental weights): in compact semisimple Lie algebras, orthogonal Cartan subalgebras always exist (where one of them can be chosen arbitrarily). Some of the consequences of this theorem are the following. $(i)$ If $L=\operatorname{Lie}(G)$ is such a Lie algebra and $C$ is any Cartan subalgebra of $L$, then the $G$-orbit of $C^{\perp}$ is all of $L$. $(ii)$ The consequence in part $(i)$ answers a question by L. Florit and W. Ziller on fatness of certain principal bundles. It also shows that in our case, the commutator map $L \times L \to L$ is open at $(0, 0)$. $(iii)$ given any regular element $x$ of $L$, there exists a regular element $y$ such that $L = [x, L] + [y,L]$ and $x$, $y$ are orthogonal. Then we generalize this result about compact semisimple Lie algebras to the class of non-Hermitian real semisimple Lie algebras having full rank. Finally, we survey some recent related results , and construct explicitly orthogonal Cartan subalgebras in $\mathfrak{su}(n)$, $\mathfrak{sp}(n)$, $\mathfrak{so}(n)$.