The Lie algebra \(\mathfrak{su}(2)\)

Previously, I noted that the three generator matrices
\[
t_x = \begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & -1 & 0
\end{pmatrix}\\
t_y = \begin{pmatrix} 0 & 0 & 1 \\0 & 0 & 0 \\
-1 & 0 & 0
\end{pmatrix}\\
t_z = \begin{pmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}
\] obey the commutation relations
\[ \begin{equation}
 [t_x, t_y ] = t_z \\
 [t_y, t_z ] = t_x \\
 [t_z, t_x ] = t_y
\label{lasu2:eq:comrel1}
\end{equation}
\] It turns out that the matrices \(t_x\), \(t_y\) and \(t_z\) may appear in a very much different form and still follow the same set of rules \eqref{lasu2:eq:comrel1}. E.g. the three complex-valued matrices
\[
\begin{equation}
t_x = \frac{i}{2}\,\begin{pmatrix}
0 & -1 \\
-1 & 0 \end{pmatrix}\\
t_y = \frac{i}{2}\,\begin{pmatrix}
0  & i \\
-i & 0  \end{pmatrix}\\
t_z = \frac{i}{2}\,\begin{pmatrix}
-1 & 0 \\0 & 1
\end{pmatrix}
\label{lasu2:eq:repre2}
\end{equation}
\] also fulfil \eqref{lasu2:eq:comrel1}. These matrices generate three "sort-of-rotations" in a two-dimensional space with complex coordinates. Visualizing (in three dimensions) points moving in two-dimensional complex space (four dimensional space with real coordinates) is too difficult for me. Here is my best-effort result:

The red crosses and blue circles mark two unit squares [(0,0), (1,0), (1,i), (0,i)], one in each of the two complex planes which constitute the two-dimensional complex space. The animations show the motion of the two squares when rotated by \(\mathbb{1}+\alpha\cdot t_x\) (bottom left), \(\mathbb{1}+\alpha\cdot t_y\) (bottom right) and \(\mathbb{1}+\alpha\cdot t_z\) (top left); \(\alpha\) denotes the (infinitesimal) rotation angle. Note that it requires an accumulated rotation of \(4\,\pi\) for the squares to return to their original position.

The Lie algebra described by the relations \eqref{lasu2:eq:comrel1} is known as \(\mathfrak{su}(2)\) - by physicists, mathematicians call it \(A_1\).