A theorem in the representation theory of Lie algebras which states that every finite-dimensional solvable Lie algebra of linear transformations over an algebraically closed field has a common eigenvector in the underlying vector space; equivalently, representations of solvable Lie algebras are triangularizable.

Principle

Solvability restricts the action sufficiently to produce invariant one-dimensional subspaces; iterative application yields a full flag of invariant subspaces and thus a basis in which the operators are upper triangular.

Demonstration

Concrete example: a solvable Lie subalgebra of gl_n(C) can be conjugated into the algebra of upper triangular matrices; in practice one finds a common eigenvector and reduces dimension inductively to triangularize the representation.

Misapplication

Using Lie's theorem over a non‑algebraically closed field, or assuming it applies to non‑solvable algebras or infinite-dimensional representations; e.g. expecting a common eigenvector over the real numbers for every solvable real Lie algebra without complexification is incorrect.

Consequence

The theorem yields triangularization of solvable representations, which implies existence of composition series, simplifies computation of characters and weights, and underlies structure results in representation theory and the study of invariant flags.

Reversal

The converse 'if a representation is triangularizable then the acting algebra is solvable' is true in the finite-dimensional context, so triangularizability and solvability are tightly related though distinct notions in broader settings.

Boundary

Requires finite-dimensionality and an algebraically closed base field (classical statement over C); it does not hold in general over arbitrary fields without modification and fails for non‑solvable Lie algebras.

Semantic Tension

Sometimes conflated with Engel's theorem or with general triangularization results; the tension lies between conditions ensuring a single common eigenvector and stronger operator nilpotence conditions that guarantee nilpotency of the algebra.

Synthesis

Lie's theorem ties solvability of a Lie algebra to the existence of invariant one‑dimensional subspaces in finite-dimensional linear representations over algebraically closed fields, producing triangular forms that expose the representation's composition structure.