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Jordan–Chevalley Decomposition in Lie Algebras
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 62 / Issue 2 / June 2019
- Published online by Cambridge University Press:
- 28 February 2019, pp. 349-354
- Print publication:
- June 2019
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- Article
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We prove that if
$\mathfrak{s}$ is a solvable Lie algebra of matrices over a field of characteristic 0 and 
$A\in \mathfrak{s}$, then the semisimple and nilpotent summands of the Jordan–Chevalley decomposition of 
$A$ belong to 
$\mathfrak{s}$ if and only if there exist 
$S,N\in \mathfrak{s}$, 
$S$ is semisimple, 
$N$ is nilpotent (not necessarily 
$[S,N]=0$) such that 
$A=S+N$.
