<s>[INST] Hopf fibration from $SO(3)$ Lie algebra generators? 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.
 [/INST] 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}
</s>