SO(8)
In mathematics, SO is the special orthogonal group acting on eight-dimensional Euclidean space. It could be either a real or complex simple [Lie group] of rank 4 and dimension 28.
Spin(8)
Like all special orthogonal groups SO with n ≥ 2, SO is not simply connected. And like all SO with n > 2, the fundamental group of SO is isomorphic to Z2. The universal cover of SO is the spin group Spin.Center
The center of SO is Z2, the diagonal matrices , while the center of Spin is Z2×Z2.Triality
SO is unique among the simple Lie groups in that its Dynkin diagram, , possesses a three-fold symmetry. This gives rise to peculiar feature of Spin known as triality. Related to this is the fact that the two spinor representations, as well as the fundamental vector representation, of Spin are all eight-dimensional. The triality automorphism of Spin lives in the outer [automorphism group] of Spin which is isomorphic to the symmetric group S3 that permutes these three representations. The automorphism group acts on the center Z2 x Z2. When one quotients Spin by one central Z2, breaking this symmetry and obtaining SO, the remaining outer automorphism group is only Z2. The triality symmetry acts again on the further quotient SO/Z2.Sometimes Spin appears naturally in an "enlarged" form, as the automorphism group of Spin, which breaks up as a semidirect product: Aut ≅ PSO ⋊ S3.
Unit octonions
Elements of SO can be described with unit octonions, analogously to how elements of SO can be described with unit complex numbers and elements of SO can be described with unit quaternions. However the relationship is more complicated, partly due to the non-associativity of the octonions. A general element in SO can be described as the product of 7 left-multiplications, 7 right-multiplications and also 7 bimultiplications by unit octonions.It can be shown that an element of SO can be constructed with bimultiplications, by first showing that pairs of reflections through the origin in 8-dimensional space correspond to pairs of bimultiplications by unit octonions. The triality automorphism of Spin described below provides similar constructions with left multiplications and right multiplications.
Octonions and triality
If and, it can be shown that this is equivalent to, meaning that without ambiguity. A triple of maps that preserve this identity, so that is called an isotopy. If the three maps of an isotopy are in, the isotopy is called an orthogonal isotopy. If, then following the above can be described as the product of bimultiplications of unit octonions, say. Let be the corresponding products of left and right multiplications by the conjugates of the same unit octonions, so,. A simple calculation shows that is an isotopy. As a result of the non-associativity of the octonions, the only other orthogonal isotopy for is. As the set of orthogonal isotopies produce a 2-to-1 cover of, they must in fact be.Multiplicative inverses of octonions are two-sided, which means that is equivalent to. This means that a given isotopy can be permuted cyclically to give two further isotopies and. This produces an order 3 outer automorphism of. This "triality" automorphism is exceptional among spin groups. There is no triality automorphism of, as for a given the corresponding maps are only uniquely determined up to sign.