  The present paper addresses open questions regarding the handling of the spin
supplementary condition within the effective field theory approach to the
post-Newtonian approximation. In particular it is shown how the covariant spin
supplementary condition can be eliminated at the level of the potential (which
is subtle in various respects) and how the dynamics can be cast into a fully
reduced Hamiltonian form. Two different methods are used and compared, one
based on the well-known Dirac bracket and the other based on an action
principle. It is discussed how the latter approach can be used to improve the
Feynman rules by formulating them in terms of reduced canonical spin variables.
