Infinitesimal rotations

This animated GIF

was created by repeatedly rotating the vertex points \((x,y,z)\) through a small angle \(\alpha\) around the x-axis into the point \((x',y',z')\),
\[
\begin{pmatrix}
x\\
y\\
z
\end{pmatrix}
\rightarrow
\begin{pmatrix}x'\\
y'\\
z'
\end{pmatrix}
=
R_x(\alpha)
\cdot
\begin{pmatrix}
x\\
y\\
z
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 & 0 \\
0 & \cos\alpha & \sin\alpha \\
0 & -\sin\alpha & \cos\alpha
\end{pmatrix}
\cdot
\begin{pmatrix}
x\\
y\\
z
\end{pmatrix}
\]
If \(\alpha \ll 1\) we may write
\[
\begin{pmatrix}
x'\\
y'\\
z'
\end{pmatrix}
\approx
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & \alpha \\
0 & -\alpha & 1
\end{pmatrix}
\cdot
\begin{pmatrix}
x\\
y\\
z
\end{pmatrix}
=
\left(\mathbb{1} + \alpha\cdot t_x\right)\cdot
\begin{pmatrix}
x\\
y\\
z
\end{pmatrix}
\]
where the matrices \(\mathbb{1}\) and \(t_x\) are defined as
\[
\mathbb{1} \equiv \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}
\\
t_x \equiv \begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & -1 & 0
\end{pmatrix}
\]
In a sense, the matrix \(t_x\) "generates" a rotation about the x-axis. Likewise, the (generator) matrices for rotations about the y- and z-axis ("y-rotation" and "z-rotation") are
\[
t_y \equiv \begin{pmatrix} 0 & 0 & 1 \\
0 & 0 & 0 \\
-1 & 0 & 0
\end{pmatrix}\\
t_z \equiv \begin{pmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}
\]
If I understand the Lie-terature correctly, the concept of infinitesimal rotations is one of the key ideas in the study of continuous groups.

Now one might ask how much "deviation from commutativity" is caused by two consecutive infinitesimal rotations? By "deviation from commutativity" I mean the distance between points infinitesimally rotated twice, if the order of the two infinitesimal rotations is exchanged. E.g., for infinitesimal x- and y-rotations we find
\[
\left(R_x \cdot R_y - R_y \cdot R_x\right)
\cdot
\begin{pmatrix}
x\\
y\\
z
\end{pmatrix}\\
\approx
\left(
\left(\mathbb{1} + \alpha\cdot t_x\right)\cdot
\left(\mathbb{1} + \alpha\cdot t_y\right)
-
\left(\mathbb{1} + \alpha\cdot t_y\right)\cdot
\left(\mathbb{1} + \alpha\cdot t_x\right)
\right)
\cdot
\begin{pmatrix}
x\\
y\\
z
\end{pmatrix}\\
=  \alpha^2\cdot\left(t_x \cdot t_y - t_y \cdot t_x\right) \cdot
\begin{pmatrix}
x\\
y\\
z
\end{pmatrix}
\]which, by actually doing the matrix multiplication, is found to be
\[
\alpha^2\cdot\left(t_x \cdot t_y - t_y \cdot t_x\right) \cdot
\begin{pmatrix}
x\\
y\\
z
\end{pmatrix}
=  \alpha^2\cdot t_z \cdot
\begin{pmatrix}
x\\
y\\
z
\end{pmatrix}
\]
With the notation
\[
[t_x,t_y] \equiv t_x \cdot t_y - t_y \cdot t_x
\]
we may write
\[
\begin{equation}
[t_x,t_y] = t_z
\label{ir:eq:comrel1}
\end{equation}
\]
and likewise
\[
\begin{equation} 
[t_y, t_z] = t_x \\ 
[t_z, t_x] = t_y
\label{ir:eq:comrel2}
\end{equation}
\]
The visual interpretation of equation \eqref{ir:eq:comrel1} is illustrated here

Eight points are rotated about the x-axes, then about the y-axis. The result (blue dots) is compared with the results obtained by the same rotations, however applied in reverse order (black dots). The difference (red line) corresponds to a rotation about the z-axis. Formally, the commutator of the two infinitesimal rotations \([t_x, t_y]\) is again an infinitesimal rotation, a z-rotation generated by \(t_z\).