  In this paper a geometric approach to the special relativity (SR) is used
that is called the "invariant special relativity" (ISR). In the ISR it is
considered that in the four-dimensional (4D) spacetime physical laws are
geometric, coordinate-free relationships between the 4D geometric,
coordinate-free quantities. It is mathematicaly proved that in the ISR the
electric and magnetic fields are properly defined vectors on the 4D spacetime.
According to the first proof the dimension of a vector field is mathematicaly
determined by the dimension of its domain. Since the electric and magnetic
fields are defined on the 4D spacetime they are properly defined 4D vectors,
the 4D geometric quantities (GQs). As shown in an axiomatic geometric
formulation of electromagnetism with only one axiom, the field equation for the
bivector field F [33], [T. Ivezi\'c, Found. Phys. Lett. 18, 401 (2005), arXiv:
physics/0412167], the primary quantity for the whole electromagnetism is the
bivector field F. The electric and magnetic fields 4D vectors E and B are
determined in a mathematically correct way in terms of F and the 4D velocity
vector v of the observer who measures E and B fields. Furthermore, the proofs
are presented that under the mathematicaly correct Lorentz transformations,
which are first derived by Minkowski and reinvented and generalized in terms of
4D GQs, e.g., in [23], [T. Ivezi\'c, Phys. Scr. 82, 055007 (2010)], the
electric field 4D vector transforms as any other 4D vector transforms, i.e.,
again to the electric field 4D vector; there is no mixing with the magnetic
field 4D vector B, as in the usual transformations (UT) of the 3D fields. This
formulation with the 4D GQs is in a true agreement with experiments in
electromagnetism, e.g., the motional emf.
