2 Corrected formatting.

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?

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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?