Ah OK, thanks for the clarification.

But a question: I seem to recall that the B field can be derived from the motion of the charges that are responsible for the E field. I wonder, then, is it possible that the transformation of E and B fields into hybrids is itself attributable to the effects of length contraction and time dilation? Or is this just woolly thinking on my part?

You are probably thinking of the case of a conduction current in a wire neutral and stationary in some 'lab frame' S. Which generates an azimuthal purely magnetic field

**B** in S. Move to a frame S' having uniform linear motion at velocity

**v** along the wire axis, then relativistic length contraction alone is enough to find that in S' the wire is no longer electrically neutral, but has a linear charge density, and ensuing radial electric field

**E**', in addition to

**B**' = γ

**B**. Owing to the differential in relativistic contractions of positive lattice ion spacings vs that for conduction electrons moving relative to the lattice. Thus a test charge q stationary in S' feels a purely electric force

**F**' = q

**E**' acting radial to the wire axis.

Transform back into S, and the (generally only slightly different) force

**F** =

**F**'/γ on q is interpreted as q moving normal to the azimuthal

**B**, thus

**F** = -q

**v**x

**B**, also acting radial to the wire axis.

That however is a special case in the sense in most situations you can't get away with simply appealing to length contraction to explain EM phenomena in general.

One approach starts with either the 3-potentials φ,

**A**, or invariant 4-potential

*A*, owing to some charge/current configuration, and apply the relativistic transforms (allowing for propagation delay) relevant to the definitions

**E** = -∇φ -∂

**A**/∂t, B = ∇x

**A**. This yields the Liénard–Wiechert potentials, which were developed before SR, but are relativistically correct:

https://en.wikipedia.org/wiki/Liénard–Wiechert_potential
Alternately, work from the fields direct and apply the usual transformations:

https://www.phy.duke.edu/~rgb/Class/Electrodynamics/Electrodynamics/node140.html
The familiar transformations start at (16.170)