PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-05

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Opposite root spaces bracket to the Killing-dual line

Statement

Let $α$ be a root and let $x \in g_{α}$ , $y \in g_{- α}$ . Then

$[x, y] = B (x, y) H_{α},$

where $H_{α}$ is the vector from The Killing-dual vector attached to a root. In particular, $[g_{α}, g_{- α}]$ is the line $C H_{α}$ .

Facts & Assumptions

Given: A root $α$ , vectors $x \in g_{α}$ , $y \in g_{- α}$ , and the Killing-dual vector $H_{α}$ .

Proof

technique · direct

1.1

By Brackets of root spaces add their roots, the bracket $[x, y]$ lies in $g_{0} = h$ . For every $h \in h$ , invariance of the Killing form gives $B ([x, y], h) = B (x, [y, h]) = α (h) B (x, y) = B (x, y) B (H_{α}, h)$ .

givenalgebra

2.1

Because both $[x, y]$ and $B (x, y) H_{α}$ lie in $h$ and have the same Killing pairings with every $h \in h$ , nondegeneracy of the restriction from The Killing form pairs only opposite root spaces forces $[x, y] = B (x, y) H_{α}$ .

step 1.1

3.1

Choosing $x$ and $y$ with $B (x, y) \neq 0$ shows that the image of $[g_{α}, g_{- α}]$ is exactly the line $C H_{α}$ .

step 2.1∎

Depends on

Used by

Dependency tree · two levels

6 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, Lie Groups and Lie Algebras I (standard reference, not scraped)