Clifford systems, Clifford structures, and their canonical differential forms

A comparison among different constructions in H-2 congruent to R-8 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 "Kahler 2-forms", entering both in quaternion Kahler 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 in R-16 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.