Classification of invariant valuations on the quaternionic plane

We describe the orbit space of the action of the group $\mathrm{Sp}(2)\mathrm{Sp}(1)$ on the real Grassmann manifolds $\mathrm{Gr}_k(\mathbb{H}^2)$ in terms of certain quaternionic matrices of Moore rank not larger than $2$. We then give a complete classification of valuations on the quaternionic plane $\mathbb{H}^2$ which are invariant under the action of the group $\mathrm{Sp}(2)\mathrm{Sp}(1)$.