Ado’s theorem

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field of characteristic $0$ (typically $\mathbb{R}$ or $\mathbb{C}$).

Theorem (Ado). There exists a finite-dimensional vector space $V$ and an injective Lie algebra homomorphism

so $\mathfrak{g}$ is isomorphic to a Lie subalgebra of the general linear Lie algebra .

Equivalently, every finite-dimensional Lie algebra is a “matrix Lie algebra” up to isomorphism.

Motivation. Ado’s theorem guarantees that Lie algebra theory can be studied inside $\mathfrak{gl}_n$ using linear algebra. It complements Lie’s third theorem , which integrates Lie algebras to (simply connected) Lie groups, by ensuring that the infinitesimal data can always be realized concretely as endomorphisms.

Remark. Given a representation $\rho$ of $\mathfrak{g}$ and a simply connected Lie group $G$ with Lie algebra $\mathfrak{g}$, $\rho$ integrates to a group representation $G\to \mathrm{GL}(V)$; this bridges Ado’s theorem with the study of linear Lie groups .