PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

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Brackets of root spaces

Statement

Let $h$ be a Cartan subalgebra of a finite-dimensional complex semisimple Lie algebra $g$ , and let $g_{α}$ be the root spaces of Root and root space for $α \in h^{*} \cup {0}$ , with $g_{γ} = 0$ whenever $γ$ is not a root or $0$ . Then $[g_{α}, g_{β}] \subseteq g_{α + β}$ for all $α, β \in h^{*}$ .

Facts & Assumptions

Given: Such $g, h$ and functionals $α, β$ .

[L1]

For $H \in h$ , the operator ${ad}_{H}$ of Derivations of Lie algebras is a derivation: ${ad}_{H} [x, y] = [{ad}_{H} x, y] + [x, {ad}_{H} y]$ (Derivations form a Lie algebra and inner derivations an ideal).

[L2]

The root spaces are the eigenspaces $g_{γ} = {x : [H, x] = γ (H) x for all H \in h}$ and the root-space decomposition holds (Root and root space, Root-space decomposition).

Proof

technique · direct

1.1L1algebra

Let $x \in g_{α}$ , $y \in g_{β}$ and $H \in h$ . By [L1], $[H, [x, y]] = [[H, x], y] + [x, [H, y]] = [α (H) x, y] + [x, β (H) y] = (α + β) (H) [x, y]$ .

2.1L2step 1.1algebra∎

Since the functional $α + β$ acts on $[x, y]$ by the scalar $(α + β) (H)$ for every $H \in h$ , step 1.1 says $[x, y] \in g_{α + β}$ whenever $α + β$ is a root or $0$ , and says $[x, y] = 0 \subseteq g_{α + β} = 0$ when $α + β$ is neither, which is the convention of the statement; this covers all $x \in g_{α}$ and $y \in g_{β}$ , so $[g_{α}, g_{β}] \subseteq g_{α + β}$ . The case $α = β = 0$ says that $h$ is a subalgebra, which it is.

Depends on

Used by

The only scalar multiples of a root that are roots are plus or minus the root Corollary
Positive and negative nilpotent subalgebras and the Borel Definition
Standard parabolic and Levi subalgebras from a subset of simple roots Definition
Root-space brackets for matrix units Example
If alpha and beta are roots then alpha plus beta is always a root False statement
Chevalley basis and real structure constants Lemma
Normality of the nilpotent cone and Springer functions Lemma
Parabolic semi-regular duality and the reverse-column label map Lemma
Residue Casimir operators and local affine-Hecke actions Lemma
Orthogonality of root spaces and nondegeneracy on the Cartan subalgebra Proposition
Root reflections are induced by inner automorphisms Proposition
The adjoint highest weight is the highest root Proposition
The bracket of opposite root spaces is the root line Proposition
Existence and uniqueness up to isomorphism of the split real form Theorem
Root spaces of a complex semisimple Lie algebra are one-dimensional Theorem
The root sl₂ triple Theorem
The root-string property Theorem
Triangular decomposition Theorem
Vogan and Satake diagrams give equivalent real form classifications Theorem
Vogan diagram for a fixed Cartan involution is well defined up to equivalence Theorem

Dependency tree · two levels

13 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

Pavel Etingof, MIT 18.745 Lie Groups and Lie Algebras I, Lectures 19–24 (standard reference, not scraped)