The Gordon decompositions of the inertial currents of the Dirac electron correspond to a Foldy-Wouthuysen transformation

We consider Dirac's free electron theory on the first quantized level. We decompose its canonical spin current \'a la Gordon and find a conserved ``Gordon spin'' current which turns out to be equivalent to the Hilgevoord-Wouthuysen spin. We can conclude therefrom that the Gordon-type decomposition mentioned above corresponds to a Foldy-Wouthuysen transformation which transforms the Dirac wave function from the conventional Dirac-Pauli to the Newton-Wigner representation.