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A recent series of works by M. Dubois-Violette, I. Todorov and S. Drenska characterised the SM gauge group as the subgroup of SO(9) that, in the octonionic model of the later, preserves the split of the space of octonions into a copy of the complex plane plus the rest. This description, however, proceeded via the exceptional Jordan algebra J_3^8 and its group of automorphisms F_4, and remained rather indirect. The goal of this paper is to provide as explicit description as possible and also clarify the underlying geometry. It is well-known that the groups SO(3), SO(5) and SO(9) have a complex, quaternionic and octonionic models respectively. The first of these is the familiar realisation of the (double cover of) the rotation group in three dimensions in terms of 2x2 special unitary matrices. Replacing complex numbers with quaternions and octonions one gets SO(5) and SO(9) respectively. Choosing a unit imaginary quaternion or octonion then equips quaternions or octonions with an almost complex structure, and thus introduces the splits H=C+C or O=C+C^3, where the first copy of C is the one containing the imaginary unit that generates the almost complex structure. The subgroup of transformations in SO(5) that preserves H=C+C is SU(2)xU(1)/Z_2. The subgroup of transformations in SO(9) that preserves the split C=C+C^3 is the Standard Model gauge group SU(3)xSU(2)xU(1)/Z_6. We explain all these statements, as well as work out their analogs for the split quaternions and octonions.

SO(9) characterization of the standard model gauge group