= Small rotation commutator
{title2=$C_\varepsilon=I+\varepsilon^2Z+O(\varepsilon^3)$}
Use the <cross-product Lie algebra> convention $X=\widehat{e_1}$, $Y=\widehat{e_2}$, $Z=\widehat{e_3}$, so $[X,Y]=Z$. Put $R_x(t)=\exp(tX)$ and $R_y(t)=\exp(tY)$, using the <matrix exponential>. The finite <group commutator> is
$$
C_\varepsilon=R_x(\varepsilon)R_y(\varepsilon)R_x(-\varepsilon)R_y(-\varepsilon)=I+\varepsilon^2Z+O(\varepsilon^3).
$$
Here $\varepsilon\to0$ and <big O notation> refers to any fixed <matrix norm>. Active <rotation matrices> act on column <vectors> from right to left: first rotate about $y$ by $-\varepsilon$, then about $x$ by $-\varepsilon$, then about $y$ by $+\varepsilon$, and finally about $x$ by $+\varepsilon$. Swapping $x$ and $y$ replaces the product by its inverse and changes the leading <Lie bracket> to $-Z$.
Writing $c=\cos\varepsilon$ and $s=\sin\varepsilon$, direct <matrix multiplication> gives
$$
C_\varepsilon e_1=\begin{pmatrix}c^2+cs^2\\s^2(2c-c^2)\\s(c^3-c^2+s^2)\end{pmatrix}=e_1+\varepsilon^2e_2+O(\varepsilon^3).
$$
For $\varepsilon=0.1$ radians, $C_\varepsilon e_1\approx(0.999950208,0.009966462,0.000501230)$, whereas $e_1+\varepsilon^2Ze_1=(1,0.01,0)$. Thus the residual is a positive rotation about $z$ to leading order; at finite $\varepsilon$ it also has smaller components. The <Lie bracket> captures the leading effect of the four moves rather than their entire finite motion. See https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec_full.pdf#page=47[Etingof, Section 7.2, Proposition 7.11].
\Image[/lie-bracket-rotation-order.png]
{title=Four small rotations leave the motion predicted by their Lie bracket}
The left panel projects the actual four arcs onto the $yz$-plane for $\varepsilon=0.25$; the start and endpoint are close but distinct. The right panel divides the residual coordinates by $\varepsilon^2$, making the limit $(\Delta y/\varepsilon^2,\Delta z/\varepsilon^2)\to(1,0)$ visible. This is a numerical picture of $[X,Y]=Z$ in the <SO(3) Lie algebra>.
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