  We diagonalize the 8-band Kane Hamiltonian with a proper inclusion of the
interband matrix elements of the scalar and vector potentials. This leads,
among other results, to a modification of the conventional expression for the
spin-orbit coupling (SOC) strength in narrow-gap semiconductors with the zinc
blende symmetry. We find that in GaAs, at low temperatures, the correct
expression for the SOC strength is actually twice as large as usually
considered. In InSb it is $1.76$ times larger. We also provide a proper
treatment of the interband matrix elements of the position operator. We show
that the velocity operator in a crystal should be defined as a time-derivative
of a fictitious position operator rather than the physical one. We compute the
expressions for both these position operators projected to the conduction band
of the 8-band Kane model. We also derive an expression for the velocity
operator and demonstrate that the SOC strength in it differs from the SOC
strength in the Hamiltonian. The ratio between them is not equal to $1$, as it
is often assumed for the Rashba model. It does not equal $2$ either. The
correct result for this ratio is given by a rational function of the parameters
of the model. This function takes values between $4(23+3\sqrt{2})/73\approx
1.49$ and $2$. Our findings modify a vast number of research results obtained
using the Rashba model and provide a path for a consistent treatment of the
latter in future applications.
