Quaternions are a useful tool for manipulating rotations in a lot of common applications. But that wasn't my point here. Also quaternions are hard to grasp by humans. I find axis-angle much more palatable in general.

Imaginary numbers can be represented with a 2x2 skew symmetric matrix with no stretch of the imagination at all. And 3x3 skew symmetric matrixes represent rotations most compactly with only 3 actual variables. Instead of 4 for quaternions, 9 for "classic rotation matrixes", or the need to tell which is the order of the angles if you're given 3 euler angles.

There are interesting applications of Lie Algebra on SO(3) [1], notably in computer vision where a global energy is minimized across two successive rgb-d "shots" in order to recover the infinitesimal rotation [2]. It's going to be easier to minimize energy on something that is most compactly defined, and always amounts to a valid rotation.

[1] https://en.wikipedia.org/wiki/Rotation_group_SO(3)#Lie_algeb... [2] https://vision.in.tum.de/_media/spezial/bib/kerl13icra.pdf