I'll now consider what happens for a Kaluza-Klein split of a 14D space-time into a "large" 4D one and a "small" 10D space.
Rotations: SO(13,1) -- like Lorentz group, SO(3,1) -- rotations/boosts
Overall: Euc(13,1) -- like Poincaré group, Euc(3,1) -- translations, rotations/boosts
Doing the split, we get SO(13,1) -> SO(3,1) * SO(10) and Euc(13,1) -> Euc(3,1) * Euc(10)
If the small space is a 10-sphere, then we get SO(11) instead of Euc(10).
This is awfully interesting, because one proposed GUT gauge group is SO(10). Let's see what its particle content is.
Particle | Space-time | Gauge
Elem. fermions | Spinor: 2+2* | Spinor: 16+16*
Higgs particles | Scalar: 1 | Vector: 10
Gauge particles | Vector: 4 | ASym 2-tensor: 45
Graviton | Sym 2-tensor: 9+1 | Scalar: 1
By chirality, the elementary fermions are (2,16) + (2*,16*)
Let's see what 14D particles can produce these particles.
Spinor: 64 -> (2,16) + (2*,16*), 64* -> (2,16*) + (2*,16)
Vector: 14 -> (4,1) + (1,10)
Sym 2-tensor: 104+1 -> (9+1,1) + (4,10) + (1,54+1)
ASym 2-tensor: 91 -> (3+3*,1) + (4,10) + (1,45)
Spinor: elementary fermions
Vector: Higgs particles + extras
Sym traceless 2-tensor: Graviton + extras
ASym 2-tensor: no gauge particles!
One needs something like an antisymmetric 3-tensor to get the gauge particles:
364 -> (4,1) + (3+3*,10) + (4,45) + (1,120)
So Eric Weinstein *may* have been able to get the Standard Model's particles out of his model.