Extension to nonrepeating part

For $\ell< \hat{\ell}$, $\mathbf{G}^{(\ell)}$ and $\mathbf{G}^{(\ell)}_{r}$ are calculated recursively from $\ell=\hat\ell-1$ to $\ell=1$ as follows:

$$\begin{eqnarray*} \mathbf{G}^{(\ell)} & = & \mbox{\boldmath {${\cal B}$}}^{(\ell... ...}_1 + 3\mathbf{G}^{(\ell + 1)}_1 \mathbf{G}^{(\ell)}_2) \right). \end{eqnarray*}$$

Note that $\mathbf{G}^{(\hat\ell)}=\mathbf{G}$ and $\mathbf{G}_r^{(\hat\ell)}=\mathbf{G}_r$, for $r=1,2,3$, are computed in the repeating part. The intuition for the above formulation is the same as in the repeating part.