On nilpotency of the derived subalgebra of a solvable Lie algebra 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?
 A: *

*Condition (i) can be removed, as already observed by Daniel.

*In positive characteristic you can find a counterexample in the book "J. Humphreys: Introduction to Lie algebras and representation theory" (Chapter 2, Section 4, page 20, Exercise 4), so condition (ii) cannot be relaxed.

*Finally, let $H={\mathbb F}x+{\mathbb F}y+{\mathbb F}z$ be the Heisenberg algebra with basis $x,y,z$, where $z$ is central in $H$ and $[x,y]=z$. Consider the ring of polynomials ${\mathbb F}[t]$ as a left $H$-module with $x$ acting as $d/dt$, $y$ acting by multiplication by $t$, and $z$ acting as the identity. Now consider the split extension $L=H\ltimes {\mathbb F}[t]$. Then $L$ is solvable of derived length 3, but the derived subalgebra $[L,L]={\mathbb F}z+{\mathbb F}[t]$ is not nilpotent. Thus condition (iii) cannot be removed.
A: Update
The bug in the example in Salvo's answer above has finally been fixed! It is now entirely correct.
My original concern was that, due to the requirement that the Jacobi Identity be satisfied, an infinite-dimensional Lie algebra (or even a finite-dimensional one) cannot be defined simply by playing with relations among basis vectors (which is what Salvo and I did initially, thus leading to a number of false starts). Hence, one needs to either work with "natural" examples of infinite-dimensional Lie algebras (as Professor Humphreys has mentioned in his comment below the wording of my question) or to struggle really hard with the Jacobi Identity in order to generate "non-natural" examples.
