nLab matrix Lie algebra

Definition

A matrix Lie algebra is the Lie algebra that is canonically a sub-Lie algebra of the general linear Lie algebra $Mat(n) = \mathfrak{gl}(n)$ of $n \times n$ matrices.

See also matrix Lie group.

Properties

By Ado’s theorem, every finite-dimensional Lie algebra over a field of characteristic zero is a matrix Lie algebra.

A similar statement fails for Lie groups. Ado’s theorem has been used as a major step in the traditional proofs of the Cartan–Lie theorem on the existence of integration of Lie algebras to Lie groups.