O}(4) \;, \label{eighteen}\end{aligned}$$ where $s^{ij}_A  = \epsilon_{ijk} s^k_A = \epsilon_{ijk}
R^k_{A a}
S^a_A$, $m^{i_1i_2 ...i_n}_A = e^{i_1}_{A a_1}e^{i_2}_{A a_2} ...
e^{i_n}_{A a_n} M_A^{a_1a_2 ... a_n}$, where $M^{A(1)}_a = dM^A_a
/ dT_A$ and where the brackets in $R^i_{A [a}
R^j_{A b]}$ mean antisymmetrization \[$u_{[a} v_{b]} \equiv
\frac{1}{2}(u_a v_b - u_b v_a)$\]. It should be emphasized that the fact that the final expression (\[eighteen\]) can be entirely written in terms of the “good” moments entering the DSX framework is far from being a trivial result. In the intermediary calculations the “bad” moments $(N_L,P_L)$ defined in Eqs. (4.22) of [@dam2] enter at several places before finally cancelling. We also remark that the last sum in curly brackets vanishes if one uses standard worldline data, as defined above.

As mentioned in Sec. I, we can get partial checks on our results by considering models where the multipole series is highly truncated. In particular, if we keep only the mass monopoles of the bodies (LD-EIH limit), formula (\[eighteen\]) reduces to $$\begin{aligned}
 m^i_{LD-EIH} &=& \sum_A  M^A z_A^i \left[1+{1 \over 2c^2}
  \left(v_A^2 - G_A^\prime\right)\right] + {\cal O}(4) \nonumber \\
  &=& \sum_A  M^A z_A^i\left[1+{1 \over 2c^2}
  \left(v_A^2 - \sum_{B \neq A} {GM^B \over r_{AB}}\right)\right]
 \nonumber \\ & & 