Ado's theorem

Remarks

The representation can be chosen so that every element of the nilradical of $\mathfrak g$ acts by a nilpotent endomorphism.

The kernel of the adjoint representation is the center of $\mathfrak g$, so the theorem is easy when the center is $0$. In general one builds a faithful finite-dimensional module of the nilradical as a quotient of its universal enveloping algebra. Then one extends the action step by step to all of $\mathfrak g$.

Example: the Heisenberg algebra with basis $x, y, z$, $[x, y] = z$ and $z$ central has nonzero center, so its adjoint representation is not faithful. It is nevertheless isomorphic to the algebra of strictly upper triangular $3 \times 3$ matrices, with $x = E_{12}$, $y = E_{23}$ and $z = E_{13}$, since $[E_{12}, E_{23}] = E_{13}$.

Finite dimension of $\mathfrak g$ is needed, since an infinite-dimensional Lie algebra cannot embed in a finite-dimensional $\mathfrak{gl}_n(k)$. The same conclusion holds in positive characteristic, by Iwasawa's theorem.

Going deeper

For a student

The theorem lets one prove results about abstract Lie algebras by working with matrices. Over $\R$, it gives a quick proof of Lie's third theorem. Every finite-dimensional real Lie algebra is the Lie algebra of a Lie group, namely of a connected subgroup of $\mathrm{GL}_n(\R)$. The group version fails: there are Lie groups, such as the universal cover of $\mathrm{SL}_2(\R)$, that have no faithful finite-dimensional representation.