Surprisingly short or elegant proofs using Lie theory 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.
 A: Many results in hyperbolic geometry are proven using the group of isometries of the hyperbolic space in an essential way.
My personal favorite is the Ergodicity of the geodesic flow$^*$.
Consider a compact hyperbolic manifold $M$.
Given a point $x\in M$, a unit vector $v\in T_xM$ and a time $t\in \mathbb{R}$,
when moving time $t$ in a unit speed along the geodesic emanating from $x$ in direction $v$, one reaches a new point $t\cdot x\in M$ pointing in direction $t\cdot v \in T_{t\cdot x}M$.
This process, which is an action of $\mathbb{R}$ on $T^1M$,  is known as the geodesic flow on $M$.
Now, the compact space $T^1M$ carries a natural volume form which gives rise to a finite measure, invariant under the $\mathbb{R}$-action. It is an important fact that this action is ergodic (in fact, mixing). This is a result of Hopf from 1939. This result was reproved by Mautner in 1957 using Lie groups reasoning. From a more modern point of view it is seen as a special application of a general result regarding vanishing at infinity of matrix coefficients of unitary representation of semi-simple Lie groups. The latter result is due to Howe-Moore and it is rather easy to prove.

*$ $ You should appreciate this theorem even if you have no interest in ergodicity per-se. It has many outside applications, e.g for counting and equidistribution results. Let me mention in particular that it is an essential ingridant in the celebrated Wise-Agol result (previously known as the virtually fibered conjecture). It enters its proof via the work of Kahn--Markovic.
A: The coefficients of the polynomial
$$(1+x)(1+x^2)(1+x^3)\cdots(1+x^n)$$
are unimodal. This innocuous-looking fact is surprisingly hard to prove, and perhaps the most elegant proof uses the representation theory of semisimple Lie algebras.  See Stanley's survey for further details and related examples.
A: I originally intended to write this as a comment but it turned out to be too long. The following is a proof of the fundamental theorem of algebra similar to OP's but which does not rely on the correspondence between connected, simply-connected Lie groups and Lie algebras but on somewhat easier results on Lie theory and some topological facts.
Suppose $K$ is a non-trivial field extension of $\Bbb{R}$ of degree $n$. Then, $K^\times$ is a connected, abelian real Lie group of dimension $n$. Commutativity implies that the exponential map $\mathrm{exp}:\Bbb{R}^n\to K^\times$ is a group morphism, and by connectedness we get that it is surjective and thus a covering map. The kernel $\Gamma$ of the exponential map is then a discrete subgroup of $\Bbb{R}^n$, isomorphic to $\Bbb{Z}^d$, and so $K^\times \simeq \Bbb{R}^{n-d}\times (S^1)^d$.
We know that $d\geq 1$ since there is torsion in $K^\times$, and in fact we must have $d=1$, for otherwise there would be $n^d$ $n$-th roots of unity in $K$. Therefore $K^\times \simeq \Bbb{R}^{n-1}\times S^1$ but also $K^\times \simeq \Bbb{R}^n\setminus\{0\}$, and we get that $n=2$ for otherwise $K^\times$ would be simply-connected.
A: A famous example is the proof of the "Hard Lefschetz theorem"
via finite-dimensional representations of $\mathfrak{sl}_2$.
For example (http://relaunch.hcm.uni-bonn.de/fileadmin/perrin/chap10.pdf):

Example 10.4.5 Let $X$ be a compact Kähler manifold of complex dimension
  $n$ (say for example a compact [smooth] projective variety). Then Hodge theory 
  defines endomorphisms $L$ and $\Lambda$ on $H^*(X,{\mathbb C})$. Set $X = L$ and
  $Y = \Lambda$ and $H(v) = (n-p)v$ for $v \in H_p(X,{\mathbb C})$.
  Then one can prove that this defines a $\mathfrak{sl}_2$-representation
  structure on $H^*(X,{\mathbb C})$. Then Corollary 10.4.4 (ii) for
  $V = H^*(X,{\mathbb C})$ is called the Hard Lefschetz Theorem.
  Of course the difficulty here is to construct the endomorphisms
  $L$ and $\Lambda$ and prove that they satisfy the correct commuting relations.

Likewise J.-P. Serre, in Complex Semisimple Lie Algebras (Springer 1966,
tr. 1987 by G. A. Jones), Remark 2 at the end of Section 5 of
"IV. The Algebra $\mathfrak{sl}_2$ and Its Representations":

Here is an example of an application of Theoremes 3 and 4, independent of 
  the interpretation of $\mathfrak{sl}_2$ as the Lie algebra of ${\rm SL}_2$:
Let $U$ be a compact Kähler variety of complex dimension $n$,
  and let $V$ be the cohomology algebra $H^*(U,{\bf C})$.  Hodge theory
  associates endomorphisms $\Lambda$ and $L$ of $V$ with the
  kählerian structure on $U$ (cf. A. Weil, Variétés kähleriennes, Chap. IV);
  let us take $X$ and $Y$ to be these endomorphisms, and define $H$ by
  the relation $Hx = (n-p)x$ if $x \in H^p(U,{\bf C})$.  Then one can check
  (Weil, loc. cit.) that $V$ becomes a $\mathfrak{g}$-module.
  By applying Theorems 3 and 4 to this module, one retrieves Hodge's theorem on
  "primitive" cohomology classes.

A: Let $\Gamma$ be an arbitrary group, and $(V, \rho), (V', \rho')$ two semisimple finite-dimensional linear representations of $\Gamma$ over a field $k$ of characteristic 0.  The tensor product representation $V \otimes V'$ is typically not irreducible when $\rho$ and $\rho'$, but is it at least semisimple?  Note that there are no "finiteness" hypotheses on $\Gamma$ at all.
The affirmative answer is a classic result due to Chevalley, the statement of which does not mention Lie theory at all, but the only known proof (as far as I'm aware) goes through applying the structure theory of linear algebraic groups to the (possibly disconnected) Zariski closure of $\Gamma$ in ${\rm{GL}}(V)$ and ${\rm{GL}}(V')$ to ultimately reduce to the semisimplicity of finite-dimensional representations of semisimple Lie algebras (i.e., those with vanishing radical) in characteristic 0.  
The proof is not short (if one is not familiar with the structure theory of linear algebraic groups), but it is very elegant and more importantly (for the question posed) it really does use Lie theory in an essential way (but Lie algebras, not Lie groups).
A: Does there exist a nonvanishing vector field on $S^3$?
Yes: $S^3$ is the unit quaternions and hence a Lie group. Thus there exist left invariant (never zero!) vector fields.
