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I think that's a good shortcut. Converting from axis-angle to quaternions is similar to switching from an angle to a vector (cos angle, sin angle). However I didn't really grok quaternions until looking at geometric algebra, and I haven't used them since then so I can't be sure my intuition is right.

yeah seeing how e1^e2 ~= i, was pretty wild for me.

This is because they map to rotors in geometric algebra which is exactly that.

https://marctenbosch.com/quaternions/

One way to look at it, is that quaternions are a bit complicated form of rotors https://www.youtube.com/watch?v=Idlv83CxP-8