[hamish_todd]

A quaternion is some amount of identity averaged with some amount of line reflection. You can visualize the line reflection as a pair of planar reflections at 90 degrees to each other. You can visualize the identity as a pair of planar reflections that are the same. Averaging the two of those will give you a pair of planar reflections that are some angle apart.

You're correct that you can construct a quaternion with the exponential map - but the most common way to make a quaternion is with a pair of vectors. Every game engine will have that function. GA will tell you how that function works - the vectors are planar reflections, you compose those to get a rotation by twice the angle, and you add (average with) the identity rotation to get a rotation by the precise angle.

> how to detach the two notions from each other

Can you say why would you want to do that? A plane always defines a planar reflection, a point always defines a point reflection, a line always defines a line reflection (assuming we're in euclidean space, which engineering is). To me this doesn't seem to be "happen to have" territory, this seems fundamental.