After careful reading and understanding of the whole paper, I believe there is a confusion here.
For the first claim, the four vectors for charge density and current density specify the laws that cover the magnetic force and electric force in relative frames according to the relative charge...
<Moderators note. Moved from the quoted thread to avoid detracting>
Newton third law does apply in magnetism, and this is proved by applying the conservation of momentum principle. This will be clearer when the causes/origin of the magnetic force is understood as described in the work "Two new...