TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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Jordan decomposition lies inside a complex semisimple Lie algebra

Statement

Assume the Axiom of Choice. Every element $x$ of a finite-dimensional complex semisimple Lie algebra $g$ has a unique abstract Jordan decomposition $x = x_{s} + x_{n}$ , and ${ad}_{x_{s}}$ , ${ad}_{x_{n}}$ are the additive Jordan–Chevalley parts of ${ad}_{x}$ .

Facts & Assumptions

Given: The Axiom of Choice, a finite-dimensional complex semisimple Lie algebra $g$ , and an element $x \in g$ .

[A1]

The Axiom of Choice is The Axiom of Choice; it is inherited through [L1] and [L2], with [L2] itself using the operator theorem [L1].

[L1]

Under the Axiom of Choice, every endomorphism of a finite-dimensional vector space over a perfect field has a unique additive Jordan–Chevalley decomposition into commuting semisimple and nilpotent parts (Over a perfect field, every endomorphism has a unique commuting semisimple-plus-nilpotent decomposition, polynomial in the endomorphism).

[L2]

Under the Axiom of Choice, if ${ad}_{x} = S + N$ is that additive decomposition, there are unique $y_{s}, y_{n} \in g$ with $S = {ad}_{y_{s}}$ and $N = {ad}_{y_{n}}$ ; they give the unique abstract Jordan decomposition of $x$ . Conversely, any abstract Jordan decomposition has adjoints $S, N$ (Jordan–Chevalley parts agree under the adjoint representation).

[L3]

An abstract Jordan decomposition is a decomposition $x = x_{s} + x_{n}$ with commuting parts whose adjoints are semisimple, respectively nilpotent (Abstract Jordan decomposition).

Proof

technique · direct

1.1

The endomorphism ${ad}_{x}$ of the finite-dimensional complex vector space $g$ has an additive Jordan–Chevalley decomposition by [L1]. Applying clause (ii) of [L2] to it produces elements $y_{s}, y_{n} \in g$ such that $x = y_{s} + y_{n}$ , $[y_{s}, y_{n}] = 0$ , ${ad}_{y_{s}}$ is semisimple and ${ad}_{y_{n}}$ is nilpotent, and such that ${ad}_{y_{s}}$ , ${ad}_{y_{n}}$ are the additive parts of ${ad}_{x}$ . By [L3] this is an abstract Jordan decomposition of $x$ .

A1L1L2L3

2.1

Let $x = u + v$ be any abstract Jordan decomposition. Clause (i) of [L2] identifies ${ad}_{u}$ and ${ad}_{v}$ with the additive Jordan–Chevalley parts of ${ad}_{x}$ , which are the parts ${ad}_{y_{s}}$ and ${ad}_{y_{n}}$ produced in step 1.1; uniqueness of the abstract decomposition in [L2] then gives $u = y_{s}$ and $v = y_{n}$ . For $g = 0$ the only element is $0$ and the decomposition $0 = 0 + 0$ satisfies the definition vacuously; the Axiom of Choice enters only through [L1] and [L2].

A1L1L2L3step 1.1∎

Depends on

Used by

Cartan subalgebras are exactly maximal toral subalgebras Theorem
Conjugacy of Cartan subalgebras Theorem
Existence of Cartan subalgebras Theorem

Dependency tree · two levels

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Sources

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