  We derive a classification of the self-adjoint extensions of the
three-dimensional Dirac-Coulomb operator in the critical regime of the Coulomb
coupling. Our approach is solely based upon the Kre{\u\i}n-Vi\v{s}ik-Birman
extension scheme, or also on Grubb's universal classification theory, as
opposite to previous works within the standard von Neumann framework. This let
the boundary condition of self-adjointness emerge, neatly and intrinsically, as
a multiplicative constraint between regular and singular part of the functions
in the domain of the extension, the multiplicative constant giving also
immediate information on the invertibility property and on the resolvent and
spectral gap of the extension.
