Triality

There is a geometrical version of triality, analogous to duality in projective geometry.

No other connected Dynkin diagram has an automorphism group of order greater than 2; for other Dn (corresponding to other even Spin groups, Spin(2n)), there is still the automorphism corresponding to switching the two half-spin representations, but these are not isomorphic to the vector representation.

Roughly speaking, symmetries of the Dynkin diagram lead to automorphisms of the Tits building associated with the group.

For Spin(8), one finds a curious phenomenon involving 1-, 2-, and 4-dimensional subspaces of 8-dimensional space, historically known as "geometric triality".

Denoting this common vector space by V, the triality may be re-expressed as a bilinear multiplication where each ei corresponds to the identity element in V. The non-degeneracy condition now implies that V is a composition algebra.