$mathrm{Spin}(9)$-invariant valuations on the octonionic plane

The dimensions of the spaces of $k$-homogeneous $mathrm{Spin}(9)$-invariant valuations on the octonionic plane are computed using results from the theory of differential forms on contact manifolds as well as octonionic geometry and representation theory. Moreover, a valuation on Riemannian manifolds of particular interest is constructed which yields, as a special case, an element of $mathrm{Val}_2^{mathrm{Spin}(9)}$.