Heisenberg Lie algebra

The three-dimensional Heisenberg Lie algebra has a basis $a,b,c$ with $[a,b]=c$ and $c$ central. It is a two-step Nilpotent Lie algebra.

On $\mathbb{C}[x]$, the operators $d/dx$, multiplication by $x$, and the identity satisfy $[d/dx,x]=1$. Sending $a,b,c$ to these operators gives a faithful irreducible infinite-dimensional representation of the Heisenberg Lie algebra.