I have a group extension $1 \to \mathbb{Z}^2 \to G \to \mathbb{Z}^2 \to 1$ with presentation $G = \langle w_1, w_2, z_1, z_2\ |\ w_1w_2w_1^{-1}w_2^{-1}, w_1z_1w_1^{-1}z_1^{-1}, w_1z_2w_1^{-1}z_2^{-1}, w_2z_1w_2^{-1}z_1^{-1}, w_2z_2w_2^{-1}z_2^{-1}, (w_1^1w_2^4)^{-1}(z_1z_2z_1^{-1}z_2^{-1})\rangle$.
This is the world's stupidest question: the group is clearly poly-(direct sum of cyclic); is it polycyclic?
Yes. A group is polycyclic if it has a subnormal series with cyclic factors. Inserting two subgroups that come from the first copy of $\mathbb{Z}$ in both factors gives you such a series.