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

Let $Q_n$ be the quasi-projective subspace of the symplectic group $\mathrm{Sp}(n)$. In this short note, we prove that the subspace Lusternik-Schnirelmann category of $Q_n$ in $\mathrm{Sp}(n)$ 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. Fern\'andez-Su\'arez, A. G\'omez-Tato and D. Tanr\'e) and has to be compared to the equality $\mathrm{cat}\,Q_{3}=3$, established by N. Iwase and T. Miyauchi.