Consult this document by Hans de Vries: 

**[The simplest, and the full derivation of Magnetism as a Relativistic side efect of ElectroStatics][1]**

He uses only **Electrostactic field and the non-simultaneity to derive the Magnetic Field**. 

It appears that the **Magnetic field is a side effect of movement of charges in the electric field**. 

If we pretend that the magnetic field exists beyond a mere artifact then the following question must be answered: 
How can the particles communicate information about its position and velocity so that, in the past, they can adjust the source of the electromagnetic energy?

(*) Hans de Vries has a very interesting online book (not yet finished) in his site.
  [1]: http://physics-quest.org/Magnetism_from_ElectroStatics_and_SR.pdf