Clifford systems, Clifford structures, and their canonical differential forms

A comparison among different constructions in \(\mathbb {H}^2 \cong {\mathbb {R}}^8\) of the quaternionic 4-form \(\Phi _{\text {Sp}(2)\text {Sp}(1)}\) and of the Cayley calibration \(\Phi _{\text {Spin}(7)}\) shows that one can start for them from the same collections of “Kähler 2-forms”, entering both in quaternion Kähler and in \(\text {Spin}(7)\) 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 in \(\mathbb {R}^{16}\) for the canonical 4-forms \(\Phi _{\text {Spin}(8)}\) and \(\Phi _{\text {Spin}(7)\text {U}(1)}\), associated with Clifford systems related with the subgroups \(\text {Spin}(8)\) and \(\text {Spin}(7)\text {U}(1)\) of \(\text {SO}(16)\). We characterize the calibrated 4-planes of the 4-forms \(\Phi _{\text {Spin}(8)}\) and \(\Phi _{\text {Spin}(7)\text {U}(1)}\), extending in two different ways the notion of Cayley 4-plane to dimension 16.