Engel's theorem

Engel's theorem states that a finite-dimensional Lie algebra of nilpotent endomorphisms has a common nonzero vector annihilated by every element. It also characterizes nilpotent Lie algebras through nilpotent adjoint actions.

Engel's lemma: A Lie algebra of nilpotent endomorphisms of a nonzero finite-dimensional vector space has a nonzero vector killed by all its elements. This representation-level result is the common-vector step underlying Engel's theorem.
Lie algebra: A vector space equipped with a bilinear, alternating bracket satisfying the Jacobi identity. Engel's theorem concerns Lie algebras represented by endomorphisms.
Friedrich Engel: A German mathematician whose work helped establish the theory of Lie algebras. Engel's name is attached to the theorem characterizing nilpotent Lie algebras.
Lie's theorem: A theorem that finite-dimensional representations of solvable Lie algebras over an algebraically closed field of characteristic zero can be simultaneously triangularized. It extends triangularization from nilpotent actions to representations of solvable Lie algebras.
Cartan's criterion: A criterion that detects solvability or semisimplicity of finite-dimensional Lie algebras through trace forms. It tests structural properties using traces rather than common annihilated vectors.
Invariant subspace: A subspace preserved by every operator in a specified family of linear maps. The common annihilated vector spans a one-dimensional invariant subspace.
Lie bracket: A bilinear operation on a Lie algebra that is alternating and satisfies the Jacobi identity. For endomorphisms, the bracket is the commutator that defines the algebra's structure.
Wilhelm Killing: A German mathematician whose work laid foundations for the classification of finite-dimensional Lie algebras. Killing's work formed part of the setting in which Engel's nilpotence results developed.
Simultaneous triangularization: The expression of a family of linear operators as upper triangular matrices in one common basis. Engel's theorem yields a strictly triangular form for its nilpotent operators.
Jordan decomposition: The decomposition of a linear operator into commuting semisimple and nilpotent parts. It separates the nilpotent component relevant to Engel's theorem from semisimple behavior.

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Ado's theoremRelated: It describes structural conditions that often simplify finite-dimensional representations.