From the triality viewpoint

Abstract

In this note we give some elementary applications of the concept of triality to the geometry of low dimensional manifolds. We exhibit explicit identifications between the compact, simply connected Lie groups, relations between principal bundles over $S⁷$, a new view of Hopf maps and expressions of the Killing- Cartan orthogonal projections from the Lie algebra of Spin (8) onto the Lie algebra of Spin (7) and $G₂$. Some basic material, from the books "Spinors and Calibrations" [H] and "Compact Projective Planes" [S-B-G-H-L-S] is reported here for the sake of completeness.