  We construct a general parametrization of the baryon octet and decuplet mass
operators including the three-body terms using the unit operator and the
symmetry-breaking factors $M^d=\textrm{diag} (0,1,0)$ and $M^s=\textrm{diag}
(0,0,1)$ in conjunction with the spin operators. Our parametrization has the
minimal number of operators needed to describe all the octet and decuplet
masses. Investigating the likely size of the three-body terms, we find that
contributions of the three-body hypercharge splittings are comparable to those
from the one- and two-body isospin splittings and that contributions of the
three-body isospin splitting operators are very small. We prove that, in
dynamical calculations, one must go to three loops to get the three-body terms.
We also find that the suggested hierarchy of sizes for terms in the most
general expression for baryon masses that involve multiple factors of $M^d$
and/or $M^s$ does not hold strictly for dynamical calculations in heavy baryon
chiral perturbation theory: terms of a given order in a meson loop expansion
may appear both with the expected factors of $M^d$ and $M^s$, and with one
factor more.
