Neighborhood of vertices in the isogeny graph of principally polarized superspecial abelian surfaces

For two supersingular elliptic curves $E$ and $E^{\mathrm{\prime }}$ defined over ${\mathbb{F}}_{p^2}$, let $[E\times E^{\mathrm{\prime }}]$ be the superspecial abelian surface with the principal polarization $\{0\}\times E^{\mathrm{\prime }}+E\times \{0\}$. We determine local structure of the vertices $[E\times E^{\mathrm{\prime }}]$ in the $(\mathrm{\ell },\mathrm{\ell })$-isogeny graph of principally polarized superspecial abelian surfaces where either $E$ or $E^{\mathrm{\prime }}$ is defined over ${\mathbb{F}}_p$. We also present a simple new proof of the main theorem in \cite{LOX20}.