PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-06

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Cartan commutator identities

Statement

With $[A, B] = A B - (- 1)^{∣ A ∣ ∣ B ∣} B A$ for homogeneous graded operators, $[L_{X}, ι_{Y}] = ι_{[X, Y]}, [L_{X}, L_{Y}] = L_{[X, Y]}, ι_{X} ι_{Y} + ι_{Y} ι_{X} = 0.$

Facts & Assumptions

Given: The manifolds, forms, vector fields, maps, and coordinates explicitly named in the statement.

[F1]

The preceding result states that For every vector field $X$ and differential form $ω$ , $L_{X} ω = d (ι_{X} ω) + ι_{X} (d ω) .$ (Cartan's magic formula).

Proof

technique · direct

1.1

Using $L_{X} = [d, ι_{X}]$ and $d^{2} = 0$ , expand the graded commutator with $ι_{Y}$ ; the antiderivation signs give $[L_{X}, ι_{Y}] = ι_{[X, Y]}$ .

F1given

2.1

The same expansion together with the bracket characterization of $L_{X}$ gives $[L_{X}, L_{Y}] = L_{[X, Y]}$ ; alternating insertion gives $ι_{X} ι_{Y} + ι_{Y} ι_{X} = 0$ .

step 1.1∎

Depends on

Used by

Lie derivative and interior product commute for all vector fields False statement

Dependency tree · two levels

17 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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)