Ado's theorem
In abstract algebra, Ado's theorem is a theorem characterizing finite-dimensional Lie algebras. Statement. Ado's theorem states that every finite-dimensional Lie algebra "L" over a field "K" of characteristic zero can be viewed as a Lie algebra of square matrices under the commutator bracket. More precisely, the theorem states that "L" has a linear representation ρ over "K", on a finite-dimensional vector space "V", that is a faithful representation, making "L" isomorphic to a subalgebra of the endomorphisms of "V". History. The theorem was proved in 1935 by Igor Dmitrievich Ado of Kazan  State University, a student of Nikolai Chebotaryov. The restriction on the characteristic was later removed by Kenkichi Iwasawa (see also the below Gerhard Hochschild paper for a proof). Implications. While for the Lie algebras associated to classical groups there is nothing new in this, the general case is a deeper result. Applied to the real Lie algebra of a Lie group "G", it does not imply that "G" has a faithful linear representation (which is not true in general), but rather that "G" always has a linear representation that is a local isomorphism with a linear group. Ado's theorem