set of Jacobi coordinates is conveniently employed. For the four-body system, the Jacobi set is identical to the K-type coordinate, and the corresponding matrix $U$ is given by $$\begin{aligned}
U_K=
\left(\begin{array}{rrrr}
1&-1&0&0\\
\frac{1}{2}&\frac{1}{2}&-1&0\\
\frac{1}{3}&\frac{1}{3}&\frac{1}{3}&-1\\
\frac{1}{4}&\frac{1}{4}&\frac{1}{4}&\frac{1}{4}\\
\end{array}\right).\end{aligned}$$ The transformation matrix for the H-type coordinate reads $$\begin{aligned}
U_H=
\left(\begin{array}{rrrr}
1&-1&0&0\\
0&0&1&-1\\
\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}\\
\frac{1}{4}&\frac{1}{4}&\frac{1}{4}&\frac{1}{4}\\
\end{array}\right).\end{aligned}$$ The K-type coordinate is obtained directly from the H-type coordinate by a transformation matrix $U_K U_H^{-1}$ $$U_KU_H^{-1}=
\left(
\begin{array}{rrrr}
1&0&0&0\\
0&-\frac{1}{2}&1&0\\
0&\frac{2}{3}&\frac{2}{3}&0\\
0&0&0&1\\
\end{array}
\right)
=\left(
\begin{array}{cc}
U_{KH}&0\\
0& 1\\
\end{array}
\right),
\label{HtoK}$$ where $U_{KH}$ is a 3$\times$3 sub-matrix of $U_K U_H^{-1}$.

Each coordinate set emphasizes particular correlations among the nucleons. As mentioned above, the H-coordinate is natural to describe the $d$+$d$ channel, whereas the K-coordinate is suited for a description of the 3$N$+$N$ partition. It is of crucial importance to include both types of motion in order to fully describe the four-nucleon dynamics [@benchmark]. In order to develop a unified method that can incorporate both types of coordinates on an equal footing, we extend the explicitly correlated Gaussian function [@suzuki00; @DGVR] to include triple global vectors $$\begin{aligned}
&&F_{L_1 L_2 (L_{12})L_3  L M}(u_1,u_2,u_3,A,{{\mbox{\boldmath $x$}}})\nonumber\\
&&\quad  ={\rm exp}\left(-{\frac{1}{2}}{\widetilde{{{\mbox{\boldmath $x$}}}}} A {{\mbox{\boldmath $x$}}}\right)
\left[[{\cal Y}_{L_1}(\widetilde{u_1}{{\mbox{\boldmath $x$}}})
 {\cal Y}_{L_2}(\widetilde{u_2}{{\mbox{\boldmath $x$}}})]_{L_{12}}
{\cal 