The generic commutation relations for the angular momentum operator are $[J_x, J_y] = i \hbar J_z$, where the $J_i$, $i = x,y,z$ are the components of the angular momentum vector operator, $\mathbf J$. The spin components obey the same commutation relations. My question is, in all physical systems using a non-relativistic model for spin, the $i$ index above always runs from 1 to 3. What is special about three?
I understand that a $3 \times 3$ rotation matrix can be represented as an element of $SO(3)$ and that spin obeys the same commutation relations as angular momentum (The reps of $SO(3)$ are objects that are appropriate for acting on the kets in the space). So, in a three dimensional system, it makes sense that there are three independent directions in which we can define a spin component. However, even for the spin 1/2 electron system, which is a two dimensional Hilbert space, we still permit these three linearly independent rotations. Why is that?