I am calculating a metric using a quaternionic approach, however, I am struggling to simplify (if possible) expressions such as:
$dq\,g \, \otimes \, d\bar{q}\, \bar{g}$
where $q$ is a general quaternionic variable and $g$ is a unit quaternion. I know that the tensor product in this case is symmetric since it comes from a coordinate, $q_1$, and using $|dq_1|^2 = dq_1 d\bar{q_1} = d\bar{q_1}dq_1$ where $q_1 = qg$
Taking $d$: $dq_1 = dq\,g+q\,dg$ and similarly $d\bar{q_1} = d\bar{q}\, \bar{g} +\bar{q}\,d\bar{g}$.
Then:
$|dq_1|^2 = dq\,g \otimes d\bar{q}\, \bar{g} + dq\,g \, \otimes \,\bar{q}\,d\bar{g} + q\, dg \,\otimes \,d\bar{q}\, \bar{g} + q\, dg \,\otimes \, \bar{q}\,d\bar{g} $
So my question is whether I can get these tensor products into forms that have, say, $dg \, \otimes \, d\bar{g}$ together.