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 $A_1$.