\sum_i (- \partial_i^2 + \omega^2 x_i^2) + a^2 \sum_{\substack{i,j \\ i\ne j}} 
       \frac{1}{x_{ij}^2} + 3b^2 \sum_{\substack{i,j \\ i\ne j}} \frac{1}{y_{ij}^2} + 2\omega \sum_i {b^{\dagger}}_i b_i \\
  & \quad + a \sum_{\substack{i,j \\ i\ne j}} \frac{1}{x_{ij}^2} [{b^{\dagger}}_i, b_i - b_j] + b 
       \sum_{\substack{i,j,k \\ i\ne j\ne k\ne i}} \frac{1}{y_{ij}^2} ([{b^{\dagger}}_i, b_i + b_j - 2b_k] \\
  & \quad - [{b^{\dagger}}_k, b_i + b_j - 2b_k]) - 3\omega (2a+2b+1), \\
  Q_{\rm CMW} & = \sum_i \left[- \partial_i + \omega x_i - a \sum_{\substack{j \\ j\ne i}} \frac{1}{x_{ij}} 
       - b \left(\sum_{\substack{j \\ j\ne i}} \frac{1}{y_{ij}} - \sum_{\substack{j,k \\ i\ne j\ne k\ne i}} 
       \frac{1}{y_{jk}}\right)\right] b_i,
\end{split}  \label{eq:super-CMW}$$ in terms of three pairs of fermionic creation and annihilation operators ${b^{\dagger}}_i$, $b_i$, $i=1$, 2, 3.

Let us now make for the latter the same kind of orthogonal transformation as that performed for the coordinates, $${b^{\dagger}}_x = \frac{1}{\sqrt{2}} ({b^{\dagger}}_1 - {b^{\dagger}}_2), \qquad {b^{\dagger}}_y =  \frac{1}{\sqrt{6}} ({b^{\dagger}}_1 + {b^{\dagger}}_2 - 2 {b^{\dagger}}_3), 
  \qquad {b^{\dagger}}_X =  \frac{1}{\sqrt{3}} ({b^{\dagger}}_1 + {b^{\dagger}}_2 + {b^{\dagger}}_3)$$ and similarly for the annihilation operators. Then, after some calculations, equation (\[eq:super-CMW\]) can be rewritten as $${\cal H}^s_{\rm CMW} = {\cal H}^s_{\rm rel} + {\cal H}^s_{\rm cm}, \qquad Q_{\rm CMW} = Q_{\rm rel} 
  + Q_{\rm cm},$$ where 