  In this paper it is proved by using the Clifford algebra formalism that the
standard transformations (ST) of the three-dimensional (3D) vectors of the
electric and magnetic fields E and B are not the Lorentz transformations (LT)
of well-defined quantities from the 4D spacetime. This difference between the
ST and the LT is obtained regardless of the used algebraic objects (1-vectors
or bivectors) for the representation of the electric and magnetic fields in the
usual observer dependent decompositions of F. The LT correctly transform the
whole 4D quantity, e.g., E_{f}=F\cdot \gamma_{0}, whereas the ST are the result
of the application of the LT only to the part of E_{f}, i.e., to F, but leaving
gamma_{0} unchanged. The new decompositions of F in terms of 4D quantities that
are defined without reference frames, i.e., the absolute quantities, are
introduced and discussed. It is shown that the LT of the 4D quantities
representing electric and magnetic fields correctly describe the motional
electromotive force (emf) for all relatively moving inertial observers, whereas
it is not the case with the ST of the 3D E and B.
