Clifford systems, Clifford structures, and their canonical differential forms

A comparison among different constructions of the quaternionic $4$-form $\Phi_{Sp(2)Sp(1)}$ and of the Cayley calibration $\Phi_{Spin(7)}$ shows that one can start for them from the same collections of "K\"ahler 2-forms", entering in dimension 8 both in quaternion K\"ahler and in $Spin(7)$ geometry. This comparison relates with the notions of even Clifford structure and of Clifford system. Going to dimension $16$, similar constructions allow to write explicit formulas for the canonical $4$-forms $\Phi_{Spin(8)}$ and $\Phi_{Spin(7)U(1)}$, associated with Clifford systems related with the subgroups $Spin(8)$ and $Spin(7)U(1)$ of $SO(16)$. We characterize the calibrated $4$-planes of the $4$-forms $\Phi_{Spin(8)}$ and $\Phi_{Spin(7)U(1)}$, extending in two different ways the notion of Cayley $4$-plane to dimension $16$.