= Solution
Let $V$ be a nonzero finite-dimensional <Irreducible Lie algebra representation>. Since $c$ is central, its action is a scalar $\lambda I$ by the <Schur lemma>. Taking the trace of $[\rho(p),\rho(q)]=\rho(c)$ gives $0=\lambda\dim V$, so $\lambda=0$.
The actions of $p,q$ now commute. Choose an <eigenvalue> $a$ of $\rho(p)$. Its nonzero <eigenspace> is preserved by $\rho(q)$, which has an <eigenvector> there, with some <eigenvalue> $b$. The span of this common <eigenvector> is invariant under all of $\mathfrak g$. Irreducibility forces it to be the whole representation. Consequently
$$
\boxed{V=\mathbb C,\qquad \rho(p)=a,\quad\rho(q)=b,\quad\rho(c)=0,
\qquad(a,b)\in\mathbb C^2.}
$$
Conversely each such assignment is a one-dimensional <Lie algebra representation> and hence irreducible. Distinct pairs give nonisomorphic representations. This classifies all finite-dimensional irreducible representations; none is faithful because all annihilate $c$.
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