U(1) rotates 2D complex values. SU(2) rotates real 3D vectors. SU(3) rotates complex 3D vectors. SU(3) combines 2D phase rotations internal to each component with 3D rotations that mix components. What, if anything, makes U(1) × SU(2) distinct from SU(3)?
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Isn't using a particular basis of 2x2 complex matrices vs using unit quaternions just a choice of representation? A real 3x3 unitary matrix rotates real 3D vectors, so I'm extrapolating to the behavior of the group of complex 3x3 matrices you get after exp(i * Gell-Mann).
Well, it can act like that but it’s a really special case (e.g. SU(18) doesn’t rotate vectors in R19). You mean SO(3) can be seen as a subgroup of SU(3)? I guess it can
