Spin(8)

In mathematics, SO(8) is the special orthogonal group acting on eight-dimensional Euclidean space. It is a real Lie group of dimension 28. Like all orthogonal groups, SO(8) is not simply connected having a fundamental group isomorphic to C₂. The universal cover of SO(8) is the spinor group Spin(8).

SO(8) is unique among the orthogonal groups in that its Dynkin diagram (shown right) possesses a high degree of symmetry. This gives rise to peculiar feature of Spin(8) known as triality. Related to this is the fact that the two spinor representations, as well as the fundamental vector representation, of Spin(8) are all eight-dimensional (In all other dimensions the spinor representation is either smaller or larger than the vector representation). The triality automorphism of Spin(8) is an outer automorphism of order three that permutes these three representations.

In other words, spin(8) can be "enlarged" to the semidirect product spin(8)S₃.

See also: Octonions, Clifford algebra, G₂