# Symbol for quaternion multiplication

I came accross different notations for the multiplication between two quaternions, e.g:

$$$$\mathbf{q}_1 \circ \mathbf{q}_2 \quad \text{or} \quad \mathbf{q}_1 \otimes \mathbf{q}_2$$$$

Which one should be preferred? Is there any standard notation for this operation ? The latter seems to be widely used for the outer product.

• The other option is no symbol between the quaternions.
– J.G.
Commented Sep 30, 2022 at 11:20
• @J.G. well if this notation is considered as the standard notation for quaternion multiplication I'm ready to accept the answer
– Gab
Commented Sep 30, 2022 at 11:34
• @ParclyTaxel for the first notation here for example, for the second one I saw it on multiple websites when searching for an answer prior to post this question
– Gab
Commented Sep 30, 2022 at 11:35
• Semanticscholar, bioRxiv, Montana State University.
– Gab
Commented Sep 30, 2022 at 11:38
• here and here, I cannot find the one on semanticscholar
– Gab
Commented Sep 30, 2022 at 11:42

$$\mathbf q_1\circ\mathbf q_2$$ recalls the interpretation of certain quaternions as rotations in 3D space, the composition of functions mapping said space to itself. $$\mathbf q_1\otimes\mathbf q_2$$ is a reminder that multiplication of quaternions is noncommutative.
But the most common and most concise notation is simply $$\mathbf q_1\mathbf q_2$$ – the quaternions being a skew-field where multiplication is defined, albeit a noncommutative one, is more than enough to merit using juxtaposition to denote multiplication. (This is also very common in group theory.)