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According to Wikipedia, a Lie algebra $\mathfrak{g}$ over a field $\mathbb{K}$ is called completely-solvable (or split solvable) if it has a chain of ideals $L_i$ such that $$0 = L_0\subset L_1\subset\cdots\subset L_=L$$ with $\dim L_i=i$.

Wikipedia also claims that a Lie algebra is completely-solvable if and only if the eigenvalues of $\mathrm{ad} X$ for all $X \in \mathfrak g$ are in the field $\mathbb{K}$.

Could someone provide a reference or a textbook for this claim?

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    In the WP article, that claim actually has a reference to Knapp, A. W. (2002): Lie groups beyond an introduction. Progress in Mathematics. 120 (2nd ed.). Boston·Basel·Berlin: Birkhäuser. It's probably a good exercise to prove it oneself too. – Torsten Schoeneberg Apr 06 '21 at 20:36

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Followint the comment of Torsten Schoenbertg, a reference can be found in Corollary 1.30 of Knapp, A. W.: Lie groups beyond an introduction. Progress in Mathematics. 140 (1996) (1st ed.).

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