Watching the Lie algebra $\mathfrak{su}(3)$ at work

Just as in the post on $\mathfrak{su}(2)$, we may attempt to visualize the effect of the $\mathfrak{su}(3)$ rotations

$$\begin{equation*} t_x = \frac{i}{2}\,\begin{pmatrix} 0 & -1 & 0\\ -1 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix} \quad t_y = \frac{i}{2}\,\begin{pmatrix} 0  & i & 0 \\ -i & 0 & 0 \\ 0 & 0 & 0  \end{pmatrix} \quad t_z = \frac{i}{2}\,\begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}\\ u_x = \frac{i}{2}\,\begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & -1\\ 0 & -1 & 0 \end{pmatrix}\quad u_y = \frac{i}{2}\,\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0  & i \\ 0 & -i & 0 \end{pmatrix}\quad u_z = \frac{i}{2}\,\begin{pmatrix} 0 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\\ v_x = \frac{i}{2}\,\begin{pmatrix} 0 & 0 & -1\\ 0 & 0 & 0\\ -1 & 0 & 0 \end{pmatrix}\quad v_y = \frac{i}{2}\,\begin{pmatrix} 0  & 0 & i \\ 0 & 0 & 0 \\ -i & 0 & 0  \end{pmatrix}\quad v_z = \frac{i}{2}\,\begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} \end{equation*}$$ As already noted $v_z = t_z+u_z$, leaving only eight linear independent generators. The resulting animation is shown in Fig. 1.

Fig. 1: The Lie algebra su(3) at work

The red, green and blue symbols mark three unit squares [(0,0), (1,0), (1,i), (0,i)], one in each of the three complex planes which constitute the three-dimensional complex space. The animations show the motion of the three squares when rotated by $T_x= \mathbb{1}+\alpha\cdot t_x$, $T_y = \mathbb{1}+\alpha\cdot t_y$, etc.; here, $\alpha$ denotes the (infinitesimal) rotation angle. Note that it requires an accumulated rotation of $4\,\pi$ for the squares to return to their original position.

When we examine the right-most three (sub-)figures of Fig. 1 it actually can be seen that $V_z$ corresponds to the combined effect of $T_z$ and $U_z$. E.g. the anti-clockwise rotation of the blue square generated by $T_z$ and its clockwise rotation generated by $U_z$ cancel out causing it to remain fixed when $V_z$ is applied.