I came accross different notations for the multiplication between two quaternions, e.g:
\begin{equation} \mathbf{q}_1 \circ \mathbf{q}_2 \quad \text{or} \quad \mathbf{q}_1 \otimes \mathbf{q}_2 \end{equation}
Which one should be preferred? Is there any standard notation for this operation ? The latter seems to be widely used for the outer product.
$\mathbf q_1\circ\mathbf q_2$ 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.)