 Solution

All brackets belong to the line $\mathbb{C}c$, and $[p,q]=c$ shows this line is exactly the first derived algebra. Since $c$ is central, the derived series of a Lie algebra isThe lower central series of a Lie algebra is likewiseFor completeness, the center of a Lie algebra is $\mathbb{C}c$: an element $ap+bq+dc$ commutes with $p$ only if $b=0$, and with $q$ only if $a=0$. The upper central series of a Lie algebra is therefore $0\subset \mathbb{C}c\subset \mathfrak{g}$, because the quotient by $\mathbb{C}c$ is abelian. Thus the Heisenberg Lie algebra is nilpotent of class two and solvable of derived length two. These properties are not mutually exclusive.