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.