  Normal-ordering provides an approach to approximate three-body forces as
effective two-body operators and it is therefore an important tool in many-body
calculations with realistic nuclear interactions. The corresponding neglect of
certain three-body terms in the normal-ordered Hamiltonian is known to
influence translational invariance, although the magnitude of this effect has
not yet been systematically quantified. In this work we study in particular the
normal-ordering two-body approximation applied to a single harmonic-oscillator
reference state. We explicate the breaking of translational invariance and
demonstrate the magnitude of the approximation error as a function of model
space parameters for $^4\rm{He}$ and $^{16}\rm{O}$ by performing full no-core
shell-model calculations with and without three-nucleon forces. We combine two
different diagnostics to better monitor the breaking of translational
invariance. While the center-of-mass effect is shown to become potentially very
large for $^4\rm{He}$, it is also shown to be much smaller for $^{16}\rm{O}$
although full convergence is not reached. These tools can be easily implemented
in studies using other many-body frameworks and bases.
