Derived series and solvable Lie algebras

Let $\mathfrak{g}$ be a Lie algebra over a field (Lie algebras over a field). For linear subspaces $A,B\subseteq \mathfrak{g}$, write

$$[A,B]=\mathrm{span}\{[a,b]:a\in A,b\in B\}\mathrm{.}$$

The derived series of $\mathfrak{g}$ is

${\mathfrak{g}}^{(0)}=\mathfrak{g},{\mathfrak{g}}^{(r+1)}=[{\mathfrak{g}}^{(r)},{\mathfrak{g}}^{(r)}](r\ge 0)\mathrm{.}$

Each ${\mathfrak{g}}^{(r)}$ is an ideal (Lie subalgebras, ideals, and center). Indeed, if $I$ is an ideal, Jacobi gives

$[x,[a,b]]=[[x,a],b]+[a,[x,b]]\in [I,I]$

for $x\in \mathfrak{g}$ and $a,b\in I$; induction starts with $I=\mathfrak{g}$. In particular the series is descending, because $[I,I]\subseteq I$ for every ideal $I$.

The Lie algebra $\mathfrak{g}$ is solvable if ${\mathfrak{g}}^{(m)}=0$ for some integer $m\ge 0$. Thus the zero Lie algebra is solvable (take $m=0$), and a nonzero abelian Lie algebra is solvable with ${\mathfrak{g}}^{(1)}=0$. No finite-dimensional or characteristic hypothesis is part of the definition.