DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Lie subalgebras, ideals, and center

Definition

Let $g$ be a Lie algebra over $k$ , and let $h$ be a linear subspace (Linear subspace of a vector space).

It is a Lie subalgebra if $[h, h] \subseteq h$ . The restricted bracket then makes $h$ a Lie algebra: bilinearity, alternation, and Jacobi are inherited from $g$ .
It is an ideal, written $i ⊴ g$ , if $[g, i] \subseteq i$ . Since the bracket is skew-symmetric, this is equivalent to $[i, g] \subseteq i$ .
The center is $Z (g) = {z \in g : [z, x] = 0 for every x \in g} .$

The center is an ideal: it is a linear subspace by bilinearity, and for $z \in Z (g)$ and $x \in g$ one has $[x, z] = 0$ , which lies in $Z (g)$ . Every ideal is therefore a Lie subalgebra, but a Lie subalgebra need not be an ideal.

Depends on

Used by

Derived series and solvable Lie algebras Definition
Free Lie algebra on a vector space Definition
Lower central series and nilpotent Lie algebras Definition
Nilradical Definition
Normalizer of a Lie subalgebra Definition
Positive and negative nilpotent subalgebras and the Borel Definition
Reductive Lie algebras Definition
Simple, semisimple, and reductive Lie algebras Definition
Solvable radical Definition
The special linear Lie algebra sl₂ Definition
Toral and maximal toral subalgebras Definition
Upper central series of a Lie algebra Definition
Cartan subalgebras of a direct sum Example
Not every Lie subalgebra is an ideal False statement
Generalized weight spaces of a nilpotent subalgebra Lemma
Orthogonal complements under invariant forms are ideals Lemma
The quotient Lie-algebra bracket is well-defined Lemma
Trace forms of faithful representations of semisimple Lie algebras are nondegenerate Lemma
A central extension of a nilpotent Lie algebra is nilpotent Proposition
A nonzero nilpotent Lie algebra has nonzero center Proposition
Derivations form a Lie algebra and inner derivations an ideal Proposition
The adjoint highest weight is the highest root Proposition
The normalizer acts on Lie algebra cohomology Proposition
Triangular decomposition Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Etingof, MIT 18.745 notes, §3.2 (standard reference, not scraped)
Kirillov, An Introduction to Lie Groups and Lie Algebras, §5.3 (standard reference, not scraped)