I am reading Andrew Steane's introduction to spinors. I understand that rotations in $\mathbb{R}^3$ correspond to SU(2) matrices in the spinor representation, which are generated by Pauli matrices as $U = \exp[i(\theta/2)\boldsymbol{n}\cdot\boldsymbol{\sigma}]$. Given a unit spinor $\boldsymbol{s} \in \mathbb{C}^2$, I can find the corresponding vector $\boldsymbol{r} \in \mathbb{R}^3$ with $\boldsymbol{r} = \boldsymbol{s}^\dagger \boldsymbol{\sigma} \boldsymbol{s}$ (Eq. 17.19). So I can plot $\boldsymbol{r}$ and see that $U$ yields a rotation.
Pauli matrices are unitary, but not SU(2). Because of that I would have expected that applying a Pauli matrix to a spinor $\boldsymbol{s}$ does not yield a rotation for $\boldsymbol{r}$. However $\sigma_1\boldsymbol{s}$ seems to be equivalent to a rotation of $\boldsymbol{r}$ about the $x$ axis, which I can represent by $U = \exp[i\pi/2\sigma_1] = i\sigma_1$.
So should Pauli matrices be considered rotations? All rotations can be expressed as SU(2) matrices, but there exist also non SU(2) matrices that correspond to rotations?
Thank you