Lie's theorem

Lie's theorem states that every finite-dimensional representation of a solvable Lie algebra over an algebraically closed field of characteristic zero can be simultaneously upper triangularized.

Common eigenvector: A nonzero vector that is an eigenvector of every operator in a specified family. Lie's theorem begins by establishing a vector shared by the representation's operators.
Lie algebra: A vector space with a bilinear, alternating bracket satisfying the Jacobi identity. The theorem concerns representations of this algebraic structure.
Engel's theorem: A theorem stating that a Lie algebra of nilpotent endomorphisms has a common nonzero vector annihilated by all of them. Engel's theorem yields strictly upper triangular actions under nilpotence, a stronger condition than solvability.
Simultaneous triangularization: The choice of one basis in which every operator in a family is upper triangular. This is the theorem's direct conclusion for the operators in a representation.
Invariant subspace: A subspace mapped into itself by every operator in a specified family. The span of a common eigenvector is invariant, enabling induction on dimension.
Solvable Lie algebra: A Lie algebra whose derived series eventually becomes zero. Solvability is the structural condition that makes simultaneous triangularization possible.
Jordan decomposition: The decomposition of a linear operator into commuting semisimple and nilpotent parts. Triangular matrices may contain both diagonalizable and nilpotent behavior, unlike a purely semisimple action.
Composition series: A finite chain of submodules whose successive quotients are simple modules. The invariant flag from Lie's theorem organizes a representation into one-dimensional quotients.
Induction: A proof method that establishes a base case and derives each case from a smaller one. Invariant subspaces let the triangularization proof pass from smaller representations to larger ones.
Lie algebra representation: A linear map from a Lie algebra to endomorphisms that preserves the Lie bracket. The theorem triangularizes the operators supplied by such a representation.

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Ado's theoremCompared with: Unlike Ado's theorem, it restricts the Lie algebras and fields under consideration.