Subspace {L}usternik-{S}chnirelmann category of quasi-projective quaternionic spaces

Let $Q_n$ be the quasi-projective subspace of the symplectic group $\mathrm{S}\mathrm{p}(n)$. In this short note, we prove that the subspace Lusternik-Schnirelmann category of $Q_n$ in $\mathrm{S}\mathrm{p}(n)$ is 2. For that, we use a quaternionic logarithm, as Singhof did in the complex case for the determination of the Lusternik-Schnirelmann category of the unitary group. Our result generalizes the known case $n=2$ (by L. Fernández-Suárez, A. Gómez-Tato and D. Tanré) and has to be compared to the equality $\mathrm{c}\mathrm{a}\mathrm{t}{Q}_3=3$, established by N. Iwase and T. Miyauchi.