  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 $16$.
