One can use the Pauli matrices $\sigma_i$ to generate $Cl_3(\mathbb{R})$ and taking commutators of these matrices gives the $SU(2)$ Lie algebra $\mathfrak{su}(2)=\biggl(\begin{matrix} ia&-z\\ z&-ia\\ \end{matrix}\biggr)$
However, one can also generate $Cl_3(\mathbb{R})$ using the $4\times4$ quaternion Cayley Matrices:$R_i=\begin{pmatrix} 0&1&0&0\\ -1&0&0&0\\ 0&0&0&-1\\ 0&0&1&0\\ \end{pmatrix}, R_j =\begin{pmatrix} 0&0&1&0\\ 0&0&0&1\\ -1&0&0&0\\ 0&-1&0&0\\ \end{pmatrix}, R_k = \begin{pmatrix} 0&0&0&1\\ 0&0&-1&0\\ 0&1&0&0\\ -1&0&0&0\\ \end{pmatrix}$
These matrices act on 4-collumn unit spinors which from what I understand are elements of Spin(3), yet the associated Lie algebra of Spin(3) is generated by the following $3\times3$ matrices:
$\pi_1=\begin{pmatrix} 0&0&0\\ 0&0&-1\\ 0&1&0\\ \end{pmatrix}, \pi_2=\begin{pmatrix} 0&0&1\\ 0&0&0\\ -1&0&0\\ \end{pmatrix}, \pi_3=\begin{pmatrix} 0&-1&0\\ 1&0&0\\ 0&0&0\\ \end{pmatrix}$
I'm stuck on how one derives these matrices from the $Cl_3(\mathbb{R})$ generators.
Moreover, in this paper: https://arxiv.org/pdf/1601.02569.pdf The author writes on page 8 that the Hopf fibration is given by the map: $\Psi \pi_i \Psi^T$ where $\Psi \in\mathbb{H}$ but how can a $3\times 3$ matrix act on a quaternion, which is a 4-column, or can be represented by a $2\times2$ or $4\times4$ matrix? This makes perfect sense to me if one replaces $\pi_i$ with $\sigma_i$ then the Hopf fibration arises if we turn the quaternions into $2\times 2$ matrices but I have no idea how $\pi_i$ act on quaternions.
The $3\times 3$ Lie algebra matrices, you have quoted, would act on the quaternion expressed as a $3\times 3$ matrix, i.e. the SO(3) group.
https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation
Rather than on a quaternion represented as a 4-column, a $2\times2$ complex matrix [SU(2)], or a $4\times4$ real matrix, as is my understanding.
A quaternion $q=a+bi+cj+dk$ is written in SO(3): \begin{equation} \Psi\;=\; \begin{pmatrix} a^2+b^2-c^2-d^2 & 2(bc-ad) & 2(bd+ac) \\ 2(bc+ad) & a^2-b^2+c^2-d^2 & 2(cd-ab) \\ 2(bd-ac) & 2(cd+ab) & a^2-b^2-c^2+d^2 \end{pmatrix} \end{equation}
The Lie algebra matrices can be used in matrix exponentiation, for example to define a $3\times 3$ quaternion (a rotation matrix) using the roll-pitch-yaw angles:
https://en.wikipedia.org/wiki/Conversion_between_quaternions_and_Euler_angles
\begin{align} \exp(\theta \pi_1)\;&=\; \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos(\theta) & -\sin(\theta) \\ 0 & \sin(\theta) & \cos(\theta) \end{pmatrix}\\ \exp(\theta \pi_2)\;&=\; \begin{pmatrix} \cos(\theta) & 0 & \sin(\theta) \\ 0 & 1 & 0 \\ -\sin(\theta) & 0 & \cos(\theta) \end{pmatrix}\\ \exp(\theta \pi_3)\;&=\; \begin{pmatrix} \cos(\theta) & -\sin(\theta) & 0\\ \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 1 \end{pmatrix} \end{align}