Isometric embeddings of half-cube graphs in half-spin Grassmannians

Let $\mathrm{\Pi }$ be a polar space of type ${\mathsf{\text{D}}}_n$. Denote by ${\mathcal G}{\delta}(\Pi)$,$\delta\in {+,-}$ the associated half-spin Grassmannians and write ${\mathrm{\Gamma }}_{\delta }(\mathrm{\Pi })$ for the corresponding half-spin Grassmann graphs. In the case when $n\ge 4$ is even, the apartments of ${\mathcal G}{\delta}(\Pi)$ will be characterized as the images of isometric embeddings of the half-cube graph $\frac{1}{2}{H}_n$ in ${\mathrm{\Gamma }}_{\delta }(\mathrm{\Pi })$. As an application, we describe all isometric embeddings of ${\mathrm{\Gamma }}_{\delta }(\mathrm{\Pi })$ in the half-spin Grassmann graphs associated to a polar space of type ${\mathsf{\text{D}}}_{n^{\mathrm{\prime }}}$ under the assumption that $n\ge 6$ is even.