= Solution
All brackets belong to the line $\mathbb Cc$, and $[p,q]=c$ shows this line is exactly the first derived algebra. Since $c$ is central, the <derived series of a Lie algebra> is
$$
\boxed{\mathfrak g^{(0)}=\mathfrak g,\qquad
\mathfrak g^{(1)}=\mathbb Cc,\qquad\mathfrak g^{(2)}=0.}
$$
The <lower central series of a Lie algebra> is likewise
$$
\boxed{\gamma_1\mathfrak g=\mathfrak g,\qquad
\gamma_2\mathfrak g=\mathbb Cc,\qquad\gamma_3\mathfrak g=0.}
$$
For completeness, the <center of a Lie algebra> is $\mathbb Cc$: 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 $0\subset\mathbb Cc\subset\mathfrak g$, because the quotient by $\mathbb Cc$ is abelian. Thus the <Heisenberg Lie algebra> is \b[nilpotent of class two and solvable of derived length two]. These properties are not mutually exclusive.
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