One more talk about the independence of Einstein's SR axioms

[Sagittarius A-Star]

Ah, yes, you're right.

I found another paper on this topic:

C.4.2 Lorentz Force

Another felicitous result is that the low-velocity transform of the electric field from lab frame to a moving frame has the same form as the lab-frame Lorentz force (per unit charge), ##F/q=E+v/c×B##, for arbitrary velocity. Of course, neither Maxwell (1873) nor Lorentz (1892) were aware of this happy result of the Lorentz transformation of 4-force (Minkowski force).

Source (see page 44):
https://physics.princeton.edu//~mcdonald/examples/maxwell_rel.pdf

via:
https://physics.princeton.edu//~mcdonald/examples/

The electromagnetic field (or wave) cannot be measured directly, but only indirectly via the acceleration of a charged particle due to the force of the local electric field in it's rest frame.

 

[Sagittarius A-Star]

At Mathpages, the Lorentz transformation of the electromagnetic field is derived from the Lorentz force and the relativistic transformation for forces:

Force Laws and Maxwell's Equations
...
Returning again to equation (1a), we see that in the absence of a gravitational field the force on a particle with q = m = 1 and velocity v at a point in space where the electric and magnetic field vectors are E and B is given by

##\ \ \ \ \ \mathbf f = \mathbf E + \mathbf v \times \mathbf B ##

...
Thus if the particle is stationary with respect to the original x,y,z,t coordinates, the force on the particle has the components

## \ \ \ \ \ f_x = E_x \ \ \ \ \ f_y = E_y \ \ \ \ \ f_z = E_z ##

...
Therefore, from equation (9), we see that the transformed components of the total electromagnetic force are
##\ \ \ \ \ f_{x'} = f_x \ \ \ \ \ f_{y'} = \sqrt{1-v^2} f_y \ \ \ \ \ f_{z'} = \sqrt{1-v^2} f_z \ \ \ \ \ \ ## (10)

...
Just as the Lorentz transformation for space and time intervals shows that those intervals are the components of a unified space-time interval, these transformation equations show that the electric and magnetic fields are components of a unified electro-magnetic field. The decomposition of the electromagnetic field into electric and magnetic components depends on the frame of reference.
...
we see that Maxwell's equations are invariant under Lorentz transformations. Moreover, any physical force consistent with special relativity must transform in accord with (10), because otherwise a comparison of the forces in different frames of reference would give different results.

Source:
https://www.mathpages.com/rr/s2-02/2-02.htm

Another related paper:

A WAY TO DISCOVER MAXWELL’S EQUATIONS THEORETICALLY

Source:
https://arxiv.org/pdf/1309.6531.pdf