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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.

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    For instance, to show that every centerless Lie group is linear (via the adjoint representation). – 2017-01-04
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    If a group has a descending central series G_0 < G_1 · · · we can construct a graded Lie ring from it. This is useful in studying free groups and nilpotent groups, but I don't know if you would consider it elementary. – 2017-01-04

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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.