  A spin projection scheme is presented which allows the decomposition of the
electric conductivity into two different spin channels within fully
relativistic $ab$ $initio$ transport calculations that account for the impact
of spin-orbit coupling. This is demonstrated by calculations of the
spin-resolved conductivity of Fe$_{1-x}$Cr$_x$ and Co$_{1-x}$Pt$_x$ disordered
alloys on the basis of the corresponding Kubo-Greenwood equation implemented
using the Korringa-Kohn-Rostoker coherent potential approximation (KKR-CPA)
band structure method. In addition, results for the residual resistivity of
diluted Ni-based alloys are presented that are compared to theoretical and
experimental ones that rely on Mott's two-current model for spin-polarized
systems. The application of the scheme to deal with the spin-orbit induced spin
Hall effect is discussed in addition.
