In his book Lie Algebra, Jacobson gives a motivation for Lie algebra as a tool used in a difficult problem in group theory - Burnside's problem.
I was wondering if there is any simple/elementary application of Lie algebra in group theory which can be illustrated in class in the beginnning/introduction while teaching the course on Lie algebra.
On
Geometric control theory. If you would like a specific example, see the Reeds-Schepp car.
The groups in control theory are the 1-parameter groups of diffeomorphisms induced by the vector fields that are the controls.
There are more elementary questions than the famous Burnside problems, namely some problems concerning the solvability class of $p$-groups. They have a proof using basic Lie algebra theory. For example, we have the following question by Burnside:
Question (Burnside 1913): What is the smallest order of a group with prime power order and derived length $k$ ?
Let us denote this order by $p^{\beta_p(k)}$. One can show the following result:
Proposition: For $k\ge 4$ and all primes $p$ we have $$ \beta_p(k)\ge 2^{k-1}+2k−4. $$ A proof only using elementary Lie algebra theory was given by L. A. Bokut in $1971$. For references on similar problems see here.
Edit: For an introductory class in Lie algebras, an easier example is perhaps to show that under certain assumptions one may put all elements of a matrix group simultaneously into triangular form. This can be proved by Lie's Theorem for Lie algebras, which is a basic result in any class on Lie algebras.