If I have a moving charge observed in frame $S$, may a Lorentz boost from $S$ in a direction not parallel to the charge's velocity in $S$ result in an electric field that has a different magnitude of the component in the direction of that boost? In other words, is a single Lorentz Boost w.r.t. a frame where a charge is moving really not a pure Lorentz boost?
And what if in frame $S$ there was another charge that is not moving? Does it mean that whether or not a Lorentz boost is pure is not really an intrinsic property of that boost, but rather dependent also on the frames in which the electric field of one or more charges are not "transformed" (i.e. the source charge's/charges' rest frame(s))?