Spin commutation relations For orbital angular momentum defined as $L= r \times p $ we can prove, in quantum mechanics, the commutation relations. Also, we could prove these relationships through the study of rotations (infinitesimal) in space. These are:
$$[L_i , L_j]=i \hbar \sum_k ε_{ijk}L_k. $$
Since there isn't an analogous definition for spin angular momentum like that of the orbital angular momentum, 


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*How can we prove the commutation relations: $$[S_i , S_j]= i \hbar \sum_k ε_{ijk}S_k. $$

*Can we follow a path similar to that of the orbital angular momentum, that is the study of rotations in some space and if yes, in what space and what would this space represent?
 A: You appear confused by how spin is introduced in ordinary QM. It is rather ad hoc:
Given a Hilbert space without spin degrees of freedom of a particle $\mathcal{H}_0$, and the spin $s$ of the particle, we take the total space of states of the particle to be $\mathcal{H}_0\otimes \mathcal{S}_s$, where $\mathcal{S}_s$ is a $2s+1$-dimensional complex Hilbert space carrying the unique irreducible representation of $\mathrm{SU}(2)$ labeled by $s$.
By construction, there are three anti-Hermitian generators $T_i\in\mathfrak{su}(2)\cong\mathfrak{so}(3)$ acting on $\mathcal{S}_s$ fulfilling the commutation relations
$$ [T_i,T_j] = \sum_k\epsilon_{ijk}T_k$$
from which you get the usual Hermitian spin operators by multiplying by $\mathrm{i}$.
For $s=1$, the space $\mathcal{S}_1$ is three-dimensional, and the action of the $T_i$ is just a real-valued rotation around the $i$-axis, but, in general, the representations of $\mathrm{SU}(2)$ are not rotations, although they may be, whenever the representation map $\mathrm{SU}(2)\to\mathrm{U}(2s+1)$ hits only the real orthogonal matrices $\mathrm{O}(2s+1)\subset\mathrm{U}(2s+1)$, which happens for integer $s$.
