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?

Infinite-dimensional Lie algebras, Noordhoff International Publ., Leyden, The Netherlands, 1974. I wrote a review for the AMS Bulletin, freely available online at www.ams.org/journals/, but don't have the book itself handy. – Jim Humphreys Dec 6 '12 at 16:40