of the traditional interpretation for the double-$\Lambda$ events.

In the present work, by noting the importance of the NAGARA data, we extend this four-body model to more general cases consisting of $\alpha+x+ \Lambda+ \Lambda$ systems with $x=n,p,d,t,^3$He and $\alpha$ ($_\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), where nuclear core parts are quite well represented by $\alpha+x$ cluster models (for example in Ref. [@Furutani80]). Here we remark that the extensive calculations are presented for the first time for $A=7-9$ double-$\Lambda$ species and that the old predictions for $_\Lambda^{ }$$_\Lambda^6$He and $_\Lambda^{ }$$_\Lambda^{\!\!\!10}$Be have been updated in a unified way. The four-body calculations are accurately performed by using the Jacobian-coordinate Gaussian-basis method of Refs. [@Kami88; @Kame89; @Hiyama95; @Hiya96; @Hiyama00; @Hiyama02; @Kamada01] with all the rearrangement channels taken into account. In our model, structure changes of nuclear cores caused by added one and two $\Lambda$ particles are treated precisely. Namely, we take into account the rearrangement effects on $\Lambda \Lambda$ bond energies induced by changes of nuclear cores. It is worthwhile to point out that the important effects of core-excitations and core-rearrangement are lacking in the frozen-core approximation used often for calculations of double-$\Lambda$ hypernuclei.

In our 