# Variational criterion for commuting tori in a compact Lie group

Let $G$ be a compact (connected) Lie group, and choose an Ad-invariant inner product $B(-,-)$ on the Lie algebra $L(G)$. I am curious about the following (local) variational criterion for two one-parameter subgroups to commute (recall that this can be detected on Lie algebras).

Proposition: Fix elements $\xi,\xi'\in L(G)$ and $g_0\in G$. Then $\xi,\operatorname{Ad}_{g_0}\xi'$ commute if and only if the function $G\to \mathbb{R}, g\mapsto B(\xi,\operatorname{Ad}_g \xi')$ has a critical point at $g=g_0$, i.e. if and only if $$\frac{d}{dt}|_{t=0} B(\xi,\operatorname{Ad}_{\exp(\eta t)g_0}\xi') = 0$$ for all $\eta\in L(G)$.

The "if" direction gives a standard Lie algebraic proof that any two maximal tori $T,T'$ are conjugate. Indeed, using surjectivity of $\exp$ for tori, take $\exp(\xi),\exp(\xi')$ to be topological generators of $T,T'$, and then by compactness take $g_0$ maximizing the function $G\to \mathbb{R}$. (We also need the classification of compact connected abelian Lie groups, cf. Adams, Theorem 2.19 and Proposition 4.14, to conclude that $T,g_0T'g_0^{-1}$ generate a torus, hence coincide by maximality.) Surjectivity of $\exp\colon L(G)\to G$ shows that $G$ is covered by tori, hence by conjugates of $T$. Reference for all of this: Ch. 16 Geodesics and Maximal Tori, Theorem 16.4 of Bump's book.

The other conjugacy proofs I know are more global, using geometric or topological ideas (e.g. computing the degree of the conjugation map $G/T\times T\to G$, or the Lefschetz number of suitable left multiplication maps like $g_0\colon G/T\to G/T$ or $g_0\colon G/NT\to G/NT$). But I have little intuition for the Proposition above, proven for $g_0=1$ using the "infintesimal adjoint action" computation \begin{align*} \frac{d}{dt}|_{t=0} B(\xi,\operatorname{Ad}_{\exp(\eta t)}\xi') = B(\xi,[\eta,\xi']) &= \frac{d}{dt}|_{t=0} B(\xi,\operatorname{Ad}_{\exp(-\xi't)}\eta) \\ &= \frac{d}{dt}|_{t=0} B(\operatorname{Ad}_{\exp(\xi't)}\xi,\eta) = B([\xi',\xi],\eta). \end{align*} (The general case immediately follows from the $g_0=1$ case.)

Question: Is there a conceptual way to understand the above Proposition, perhaps using more geometry, topology, or representation theory? Alternatively, why should one expect a variational criterion for a conjugate of one given torus to commute with another given torus?

For example, I think one can prove the "only if" direction as follows. Suppose $\exp(\mathbb{R}\xi),\exp(\mathbb{R}\xi')$ lie in a common maximal torus $T$. Using elementary representation theory of $T$, take a real $T$-invariant decomposition $L(G) = L(T)\oplus L(T)^\perp$ orthogonal with respect to $B$. Then $[\eta,\xi'] = 0$ trivially for $\eta\in L(T)$, while $\operatorname{Ad}_{\exp(-\xi't)}\eta \in L(T)^\perp$ implies $B(\xi,[\eta,\xi']) = 0$ for $\eta\in L(T)^\perp$; linearity then implies $B(\xi,[\eta,\xi']) = 0$ for all $\eta\in L(G)$.

To me, this proof of the "only if" direction is a little more conceptual (e.g. involving less computation), though I would still be happy if it could be improved further, perhaps using the principal $T$-bundle $G\to G/T$. The "only if" direction also motivates the "if" direction. But I suspect I am missing some insight behind the Proposition above, especially the "if" direction.