Of all simple Lie groups, Spin(8) has the most symmetrical Dynkin diagram. The diagram has four nodes with one node located at the center, and the other three attached symmetrically. The symmetry group of the diagram is the symmetric group S3 which acts by permuting the three legs. This gives rise to an S3 group of outer automorphisms of Spin(8). This automorphism group permutes the three 8-dimensional irreducible representations of Spin(8); these being the vector representation and two chiral spin representations. These automorphisms do not project to automorphisms of SO(8).
Roughly speaking, symmetries of the Dynkin diagram lead to automorphisms of the Bruhat-Tits building associated with the group. For special linear groups, one obtains projective duality. For Spin(8), one finds a curious phenomenon involving 1, 2, and 4 dimensional subspaces of 8-dimensional space, historically known as "geometric triality".
The exceptional 3-fold symmetry of the diagram also gives rise to the Steinberg group .
Similarly, a triality between three vector spaces over a field F is a nondegenerate trilinear map
By choosing vectors ei in each Vi on which the trilinear map evaluates to 1, we find that the three vector spaces are all isomorphic to each other, and to their duals. Denoting this common vector space by V, the triality may be reexpressed as a bilinear multiplication
Conversely, the normed division algebras immediately give rise to trialities by taking each Vi equal to the division algebra, and using the inner product on the algebra to dualize the multiplication into a trilinear form.
An alternative construction of trialities uses spinors in dimensions 1, 2, 4 and 8. The eight dimensional case corresponds to the triality property of Spin(8).