  Following an idea of Dadok, Harvey and Lawson, we apply the triality property
of SO(8) to study the comass of certain self-dual 4-forms on R^8. In
particular, we prove that the Cayley 4-form has comass 1 and that any self-dual
4-form realizing the maximal Wirtinger ratio is SO(8)-conjugate to the Cayley
4-form. We also use triality to prove that the stabilizer in SO(8) of the
Cayley form is Spin(7). The results have applications in systolic geometry,
calibrated geometry, and Spin(7) manifolds.
