model and $\alpha +\alpha+\Lambda+\Lambda$ four-body model, respectively. In the same manner, we study in this work the double-$\Lambda$ hypernuclei $_\Lambda^{ }$$_\Lambda^7$He, $_\Lambda^{ }$$_\Lambda^7$Li, $_\Lambda^{ }$$_\Lambda^8$Li, $_\Lambda^{ }$$_\Lambda^9$Li, $_\Lambda^{ }$$_\Lambda^9$Be and $_\Lambda^{ }$$_\Lambda^{\!\!\!10}$Be on the basis of the $\alpha+x+\Lambda+\Lambda$ four-body model with $x=n, p, d, t, ^3$He and $\alpha$, respectively. The $d, t (^3{\rm He})$ and $\alpha$ clusters are assumed to be inert having the $(0s)^2$, $(0s)^3$ and $(0s)^4$ shell-model configurations, respectively, and are denoted by $\Phi_s(x)$ with spin $s\, (=1, \frac{1}{2}$ or 0, respectively).

All the nine sets of the Jacobian coordinates of the four-body systems are illustrated in Fig. 1 in which we further take into account the antisymmetrization between two $\Lambda$ particles and the symmetrization between two $\alpha$ clusters when $x=\alpha$. The total Hamiltonian and the Schrödinger equation are given by $$\begin{aligned}
     H=T+\sum_{(a,b)}V_{a b}
      +V_{\rm Pauli} \; ,  \\
%(2.1)
%\end{equation}
%\begin{equation}
    ( H - E ) \, \Psi_{JM}(^A_{\Lambda \Lambda}{\rm Z})  = 0 \;,
%(2.2)\end{aligned}$$ where $T$ is the kinetic-energy operator and $V_{ab}$ is the interaction between the constituent particle-pair $a$ and $b$. The Pauli principle between the nucleons belonging to $\alpha$ and $x$ clusters is taken into account by the Pauli projection operator $V_{\rm Pauli}$ which is 