[hamish_todd]

The mixed elements are the important ones!

A quaternion with w=1, x,y,z=0 is just the identity.

A quaternion with w=0, x=1, or perhaps w=0, x=y=0.7, those would only ever be rotations by 180 degrees.

If you want arbitrary rotations, you need some combination of the two: "a little bit of 180 around this line, and a little bit of 0deg rotation/identity". That's what it means to have scalar and bivector.

If you "being careful" with wedge and inner to avoid mixtures you are doing it wrong. Geometric product is the boss, and makes excellent mixtures!

[HelloNurse]

These "mixed" elements are natural types and special cases of different objects and operations that are usually confused (e.g. "vectors" of three numbers to represent points and actual 3D vectors) or contorted (e.g. quaternions to represent rotations) or idiosyncratic (e.g. 3D cross product) in more traditional approaches.