  For a Spin(9)-structure on a Riemannian manifold M^16 we write explicitly the
matrix psi of its K\"ahler 2-forms and the canonical 8-form Phi. We then prove
that Phi coincides up to a constant with the fourth coefficient of the
characteristic polynomial of psi. This is inspired by lower dimensional
situations, related to Hopf fibrations and to Spin(7). As applications,
formulas are deduced for Pontrjagin classes and integrals of Phi and Phi^2 in
the special case of holonomy Spin(9).
