= Solution
Write $V=\sum_af_aZ^{(a)}$ and $W=\sum_bg_bZ^{(b)}$. Relation (2) reduces $[V,W]$ to brackets of individual summands. Applying relations (3), (1), and (3), in that order, gives
$$
[f_aZ^{(a)},g_bZ^{(b)}]=-g_bZ^{(b)}(f_a)Z^{(a)}-f_ag_b[Z^{(b)},Z^{(a)}]+f_aZ^{(a)}(g_b)Z^{(b)}.
$$
The middle term vanishes because the $Z^{(a)}$ commute, and every remaining term lies in their smooth span. Therefore
$$
\boxed{[V,W]\in\operatorname{span}_{C^\infty}\{Z^{(1)},\ldots,Z^{(n)}\}.}
$$
This is the involutivity condition in the <Frobenius theorem>.
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