.Continuation of a discussion in the 'electron-assisted fusion thread' At the request of @Bruce-H. Alan.
.....Thanks for the answer.
I am uncomfortable with the scheme for the right-hand-rule/cross-product listed in my last post. This scheme seems to depend on an ordering of the spatial dimensions such that, for instance, (X1, X2)-> X3 rather than X4. But the labelling of such axes with indices should be is arbitrary and I should be able to swap the labels on X3 and X4 without affecting the cross product. Yet it is, indeed, affected. Swapping the X3 and X4 labels gives (X1, X2)-> X4, which is not part of the scheme I listed above.
In brief, my objection is that you can't have a cyclic scheme such as you suggest unless you have an a priori ordering of spatial directions ... and I don't see where such an ordering comes from. This is why you can't have a right-hand rule in 4-space as far as I know.
To get back to Figure 1 of your ResearchGate paper once again, what direction does the magnetic field point for the current loop in the x,y plane? Is it in the u direction? The v direction?