Some curiosities on $\mathrm{Spin}(9)$ and the sphere $S^{15}$

Although holonomy $\mathrm{Spin}(9)$ is only possible for the two $16$-dimensional symmetric spaces $\mathbb OP^2$ and $\mathbb OH^2$, weakened holonomy $\mathrm{Spin}(9)$ conditions have been proposed and studied, in particular by Th. Friedrich. A basic problem is to have a simple algebraic formula for the canonical $8$-form $\Phi_{\mathrm{Spin}(9)}$, similar to the usual definition of the quaternionic $4$-form $\Phi_{\mathrm{Sp}(n)\cdot\mathrm{Sp}(1)}=\omega_I^2+\omega_J^2+\omega_K^2$, witten in terms of local compatible almost hypercomplex structures $(I,J,K)$.
In the talk, a simple formula for $\Phi_{\mathrm{Spin}(9)}$ is presented, discussing a family of local almost hypercomplex structures associated with a $\mathrm{Spin}(9)$-manifold $M^{16}$. Some of these complex structures, now on model spaces $\mathbb R^{16^q}$, are then used to give an approach through $\mathrm{Spin}(9)$ to the very classical problem of writing down a maximal system of tangent vector fields on spheres $S^{N-1}\subset\mathbb R^N$. If time permits, some properties of manifolds equipped with a locally conformal parallel $\mathrm{Spin}(9)$ metric will be also discussed.