In another Physics stack exchange thread here, Spin matrix for various spacetime fields I obtained the generator of rotations of the SO(2) rotation group for an infinitesimal rotation of 2D vectors. I then tried to relate this to the spin-1/2 electron system, but it appears vectors representing states for that system transform under the Pauli matrices instead. I believe this is because those sets of matrices, when scaled properly, yield the correct eigenvalues for the operators.
However, I also noticed that $$D(\omega) = \text{Id} + \omega \begin{pmatrix} 0&1\\-1&0 \end{pmatrix} = \text{Id} + i \omega \begin{pmatrix} 0&-i \\i&0 \end{pmatrix},$$ so I seemed to have made contact with one of the Pauli matrices, (i.e the scaled form of which the spin states do transform). It looks like the above is the infinitesimal version of $\exp(i\omega \hat n \cdot \sigma)$ with $$\hat n \cdot \sigma = \sigma_2 = \begin{pmatrix} 0&-i \\ i&0 \end{pmatrix}$$ which seems to mean $\hat n = (0,1,0)$. In particular, c.f Sakurai P.159, eqn (3.2.3), he says that the physical spin 1/2 electron system does transform under the operator $D(\phi) = \exp(-iS_z\phi/\hbar)$ which is analogous to what I have above with the replacements $\phi \rightarrow \omega$ and $ S_z/\hbar \rightarrow \hat n \cdot \sigma$.
So is there really a connection to the matrix I derived and the spin 1/2 electron system? I do actually think not, since the matrix I derived in that other thread came from analyzing the rotation of vectors in space-time but the spin states live in another space. On the other hand, the analysis above makes me think otherwise at the same time.
Many thanks for clarification!