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Using Engel's Theorem and Lie's Theorem, one can easily establish the following result:

Let $ \frak{g} $ be a finite-dimensional Lie algebra over an algebraically closed field $ \mathbb{F} $ of characteristic $ 0 $. If $ \frak{g} $ is solvable, then $ [\frak{g},\frak{g}] {\frak{g}},{\frak{g}}] $ is nilpotent.

In order to apply the two theorems stated at the beginning, one must assume that (i) $ \mathbb{F} $ is algebraically closed, (ii) $ \mathbb{F} $ has characteristic $ 0 $, and (iii) $ \frak{g} $ is finite-dimensional. If we relax each of these three conditions in turn, are there certain well-known counterexamples?
show/hide this revision's text 1

A certain theorem about finite-dimensional Lie algebras over an algebraically closed field with zero characteristic.

Using Engel's Theorem and Lie's Theorem, one can easily establish the following result:

Let $ \frak{g} $ be a finite-dimensional Lie algebra over an algebraically closed field $ \mathbb{F} $ of characteristic $ 0 $. If $ \frak{g} $ is solvable, then $ [\frak{g},\frak{g}] $ is nilpotent.

In order to apply the two theorems stated at the beginning, one must assume that (i) $ \mathbb{F} $ is algebraically closed, (ii) $ \mathbb{F} $ has characteristic $ 0 $, and (iii) $ \frak{g} $ is finite-dimensional. If we relax each of these three conditions in turn, are there certain well-known counterexamples?