Proposition 1

Subalgebras and quotient algebras of a solvable Lie algebra are solvable. Every extension of a solvable algebra by a solvable algebra is solvable. Every finite product of solvable algebras is solvable.

Let $g$ be a Lie algebra, $g'$ a subalgebra, $h$ an ideal of $g$, $t = g/h$ and $\phi$ the canonical mapping of $g$ onto $t$. If $g$ is solvable then $D^k g = \{0\}$ for some integer $k$, hence $D^k g' \subset D^k g = \{0\}$, $D^k t = \phi(D^k g) = \{0\}$ and hence $g'$ and $t$ are solvable. If $h$ and $t$ are solvable there exist integers $s, t$ such that

then $D^t g \subset h$, hence $D^{s+t} g = D^s(D^t g) \subset D^s h = \{0\}$ and $g$ is solvable. The last assertion follows from the second by induction on the number of factors.