Introduction to Lie Algebras and Representation Theory
《Introduction to Lie Algebras and Representation Theory》 · 第 17 页 · Chapter II，Section 4.2「Jordan-Chevalley decomposition」，Proposition
Proposition. Let $V$ be a finite dimensional vector space over $F$, $x in operatorname{End} V$. (a) There exist unique $x_s, x_n in operatorname{End} V$ satisfying the conditions: $x = x_s + x_n$, $x_s$ is semisimple, $x_n$ is nilpotent, $x_s$ and $x_n$ commute. […] The decomposition $x = x_s + x_n$ is called the (additive) Jordan-Chevalley decomposition of $x$, or just the Jordan decomposition; $x_s, x_n$ are called (respectively) the semisimple part and the nilpotent part of $x$.