Affine Lie algebra of the line
= Affine Lie algebra of the line
{title2=$[h,e]=e$}
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.