  We answer a question posed by Cornulier, Fisher, and Kashyap, proving that
for a closed cocompact subgroup $\Gamma$ of $\text{Isom}(\text{DL}(n, n))$ not
contained in $\text{Isom}^+(\text{DL}(n, n))$, the sequence $1 \to H \to \Gamma
\to D_\infty \to 1$ splits, where $H$ is the unique open normal subgroup such
that $\Gamma/H \cong D_\infty$.
