Today, I was listening to someone give an *exhausting* proof of the fundamental theorem of algebra when I recalled that there was a short proof using Lie theory:

A finite extension $K$ of $\mathbb{C}$ forms a finite-dimensional vector space over $\mathbb{C}$, so the group of units $K^\times$ would be of the form $\mathbb{C}^n\setminus\{0\}$, which is simply connected for $n>1$. Since the operation on $K^\times$ is essentially just multiplication of polynomials over $\mathbb{C}$, it must be a Lie group. In sum, if $n>1$, then $K^\times$ is a simply connected abelian Lie group, thus isomorphic (as a Lie group) to $\mathbb{C}^n$, which is absurd (since $\mathbb{C}^n$ is torsion-free). Thus, $n=1$.

What other examples are there of theorems which yield such short or elegant proofs by appealing to Lie theory?

To clarify the criteria: I'm looking for (nontrivial) theorems that are usually stated in terms outside of Lie theory (e.g. the fundamental theorem of algebra) that can be proven in a particularly short or elegant way using Lie groups or Lie algebras.