\begin{array}{ccc}
L_a&S_a&I_a\\
L_b&S_b&I_b\\
L_{ab}&S&I\\
\end{array}\right]
(-1)^{L_{ab}+J-I-L}U(S L_{ab} J \ell; IL)\nonumber\\
&&\times
{\cal A}\left[
\psi_{L_a L_b (L_{ab}) \ell L}^{(\rm space)}
\left[\psi^{(\rm spin)}_{S_a} \psi^{(\rm spin)}_{S_b}\right]_{S}
\right]_{JM}
\psi^{(\rm{isospin})}_{T_a M_{T_a}}\psi^{(\rm{isospin})}_{T_b M_{T_b}}\end{aligned}$$ with $$\begin{aligned}
\psi_{L_a L_b (L_{ab}) \ell L}^{(\rm space)}
=\left[\left[\psi^{(\rm space)}_{L_a}\psi^{(\rm space)}_{L_b}\right]_{L_{ab}}
\chi_\alpha({{\mbox{\boldmath $\rho$}}}_\alpha)\right]_L,
\label{space.part}\end{aligned}$$ where $U$ and $[ \ \ ]$ are Racah and 9$j$ coefficients in unitary form [@book].

The evaluation of the matrix element can be done in the spatial, spin, and isospin parts separately. The spin and isospin parts are obtained straightforwardly. In the following we concentrate on the spatial matrix element. The spatial part (\[space.part\]) of the total wave function is given as a product of the cluster intrinsic parts and their relative motion part. The coordinates used to describe the $2N$+$2N$ channel are depicted in Fig. \[fig.1\](a) with ${{\mbox{\boldmath $\rho$}}}_\alpha={{\mbox{\boldmath $x$}}}_3$, whereas the coordinates suitable for the $t$+$p$ and $h$+$n$ channels are shown in Fig. \[fig.1\](b) with ${{\mbox{\boldmath $\rho$}}}_\alpha={{\mbox{\boldmath $x$}}}_3'$. These coordinate sets are often called H-type and K-type. Therefore the calculation of the spatial matrix element requires a coordinate transformation involving the angular momenta $L_a, L_b,
\ell$ and $L_{ab}$. Moreover the permutation operator in ${\cal A}$ causes a complicated coordinate transformation. All these complexities are treated elegantly 