Affine Lie algebra of the line ($[h,e]=e$)

The two-dimensional Lie algebra of infinitesimal dilations and translations of a line has a basis $h,e$ with $[h,e]=e$. Its derived algebra is the abelian line spanned by $e$, but every nontrivial term of its Lower central series of a Lie algebra is that same line. It is solvable and not nilpotent.