Is there a name in Geometric Algebra for the operation on two rotors $R_1, R_2: R_1 \mapsto R_2 R_1 R_2^\dagger$?

Question

In Quaternion literature, the operation of 'conjugation' of quaternion $p$ by quaternion $q$ is defined as :

$$p \mapsto q p q^{-1}$$

In Geometric algebra, there is an analogous operation with rotors and vectors. First, in order to rotate a vector $v$ by rotor $R$, we define rotation as

$$v \mapsto R v R^\dagger$$

where $R^\dagger$ is the reverse of $R$. Next, we may apply this same operation to another rotor, namely, for two rotors $R_1, R_2$:

$$R_1 \mapsto R_2 R_1 R_2^\dagger$$

Is there a name for this operation? Is it also referred to as just 'rotation'? This seems a little confusing to me, as if we want to compose two rotations, we simply left-multiply $R_1$ by $R_2$, and composition is the intuitive meaning of 'rotating a rotor' to me. By contrast, the operator in question does something more like 'changing the basis' of the original rotor, which isn't equivalent.

Answer 1

According to this Wikipedia article, though it lacks proper citation, such an operation has been referred to by some authors as a 'versor product'.

Answer 2

The term you're looking for is "conjugation" which is not the same as "conjugate" but is a term taken from group theory (see wiki).

Anecdotally, the "versor product" is a little less used from my experience, but I've also used the word "application" in reference to applying the group action of a rotor/motor/vector/etc.