Why $2$-dimensional Lie space $L$ is always solvable?

Question

Why 2-dimensional Lie space $L$ is always solvable? For 1-dimensional it is trivial, because $[ax,by]=ab[x,x]=0$ when $a,b \in \Bbb F$.

Answer 1

For dimension $2$ it is also trivial, because we have only one Lie bracket between basis vectors $e_1$ and $e_2$, which we may chose arbitrarily non-zero, e.g. $[e_1,e_2]=e_1$. So the commutator ideal is $1$-dimensional, hence abelian, so that the Lie algebra is metabelian, i.e., solvable.