Iwasawa's theorem on Lie algebras

Remarks

Idea of the proof: in characteristic $p$, the identity $(\operatorname{ad} x)^{p} = \operatorname{ad}(x^{p})$ holds in the universal enveloping algebra $U(\mathfrak g)$. For each $x$ in a basis of $\mathfrak g$, the endomorphism $\operatorname{ad} x$ of $\mathfrak g$ is killed by a nonzero polynomial whose exponents are all powers of $p$. Evaluating this polynomial at $x$ gives a central element of $U(\mathfrak g)$. Dividing $U(\mathfrak g)$ by the ideal generated by suitable such central elements gives a finite-dimensional algebra on which $\mathfrak g$ acts faithfully by left multiplication.

Together with Ado's theorem in characteristic $0$, this shows that every finite-dimensional Lie algebra over any field has a faithful finite-dimensional representation. The adjoint representation alone does not suffice when the center is nonzero, since its kernel is the center.

Going deeper

For a student

In positive characteristic the proof is easier than in characteristic $0$. The reason is that the universal enveloping algebra has a large center, generated by $p$-th power type elements, and it is a finite module over that center. So it has many finite-dimensional quotients, and one of them carries a faithful action of $\mathfrak g$. This is part of the theory of restricted Lie algebras and $p$-polynomials.