Why do actions become commutative as we reduce the 'size' of it? If we have a physical object, then infinitesimal rotations have commutative behaviour while finite rotations have anti-commutative behaviour. I find this very intriguing as to why as we take smaller and smaller actions the actions start becomes commutative whilst for large actions it's not. So, due to this phenomena rotations become vectors as when we take differential rotations.
Reference from a physics book 'Fundamentals of physics' by Resnick, Halliday walker:
This is true not just of rotations but any Lie group actions.  'Small' actions, meaning those that are near the identity, can be written as exponentials, $R=e^{X}$. For example, for rotations, $X$ is a skew-symmetric matrix.
The Baker-Campbell-Hausdorff formula states $$e^{X}e^{Y}=e^{X+Y+[X,Y]/2+\cdots}=e^{Y}e^{X}E $$ If $X$, $Y$ are small then $E=e^{[X,Y]+\cdots}$ is almost the identity, because it consists of second order terms, so the actions related to $X$, $Y$, almost commute.
This is because small rotations are relatively close to being the identity function which commutes with everything. Lets look at two dimensional rotations to get a feel for why this is.
A rotation matrix with respect to the canonical basis is of the form $\pmatrix{\cos \theta &amp; -\sin \theta \\ \sin \theta &amp; \cos \theta} = \cos \theta I + \sin \theta J$ with $J=\pmatrix{0 &amp; -1 \\ 1 &amp; 0}$. In this case, because $\theta$ is small, $\sin \theta$ is approximately $x$ near zero, the contribution of $J$ to the rotation is also small. We can also see that $\cos \theta$ should be close to $1$ when $\theta$ is near zero so the rotation as a whole is approximately $I$.