On the explicit expressions of the canonical 8-form on Riemannian manifolds with \(\mathrm {Spin}(9)\) holonomy

Two explicit expressions of the canonical 8-form on a Riemannian manifold with holonomy group \(\mathrm {Spin}(9)\) have been given: one by the present authors and another by Parton and Piccinni. The relation between these two expressions is obtained. Moreover, it is shown that they are different only from a combinatorial viewpoint.