@pschwahn - In our previous conversation I conjectured a representation of SO(3) on the imaginary octonions, and by now I've checked it really is a representation. Once we can know it's irreducible, we know it's the spin-3 irrep.
To get this rep we start by picking a basic triple of imaginary octonions, say i, j, ℓ. They span a 3d space. Any rotation of this 3d space rotates our basic triple to some other basic triple. This gives an automorphism of the octonions and thus a transformation of the space of imaginary octonions. So we get a rep of SO(3) on the 7d space of imaginary octonions! That's all there is to it.
We could either prove this is irreducible, or check by computation that it's the spin-3 rep. I can explain more explicitly how rotating our basic triple defines an automorphism of the octonions. Showing this gives the spin-3 rep may be easier at the Lie algebra level.