The additive Jordan–Chevalley supplier

The operator theorem used by the Jordan-decomposition items on this page is Over a perfect field, every endomorphism has a unique commuting semisimple-plus-nilpotent decomposition, polynomial in the endomorphism: over a perfect field every endomorphism $T$ of a finite-dimensional vector space has a unique commuting semisimple-plus-nilpotent decomposition $T=T_s+T_n$, and both parts are polynomials in $T$. Its published contract assumes the Axiom of Choice (The Axiom of Choice); every use of it below inherits that assumption, and the items that use it declare the dependence explicitly.

On this page the theorem is applied only in the following special case: the field is $\mathbb{C}$, which is perfect, and $T={\mathrm{ad}}_x$ for an element $x$ of a finite-dimensional complex Lie algebra $\mathfrak{g}$. The theorem then produces ${\mathrm{ad}}_x=S+N$ with $S$ semisimple, $N$ nilpotent, $SN=NS$, and both $S$ and $N$ polynomials in ${\mathrm{ad}}_x$. No further operator theory is imported: the passage from these operator parts to elements of $\mathfrak{g}$ is the content of Jordan–Chevalley parts agree under the adjoint representation and Jordan decomposition lies inside a complex semisimple Lie algebra.

The cited theorem is published with the Axiom of Choice in its statement but without The Axiom of Choice in its published dependency list. That metadata defect is recorded for the canonical published-defect ledger; it does not block this page, because the assumption is declared here and propagated through every consumer.