model, it is possible to determine the $\alpha x$ and $\Lambda x$ interactions so as to reproduce all the existing binding energies of subsystems ($\alpha+x$, $x+\Lambda$, $\alpha+x+ \Lambda$ and $\alpha+\Lambda+\Lambda$) in an $\alpha+x+\Lambda+\Lambda$ system, where that of $\alpha+\Lambda+\Lambda$ is given by the NAGARA event. This feature is important to discuss the energy levels of the double-$\Lambda$ hypernuclei and to extract the $\Lambda \Lambda$ interactions because the ambiguities of $NN$ and $\Lambda N$ effective interactions are renormalized by fitting the known binding energies of subsystems phenomenologically. Our analysis is performed systematically for ground and bound excited states of the series of $\alpha+x+\Lambda+\Lambda$ systems with no more adjustable parameters in this stage, so that these predictions offer an important guidance to interpret coming double-$\Lambda$ events in the experiments and then to determine the level structure and the $\Lambda \Lambda$ interaction unambiguously.

In Section II, the calculational method with microscopic $\alpha+x+ \Lambda + \Lambda$ four-body model is described. In Section III, the interactions are introduced. Calculated results are presented and discussed in Section IV. Summary is given in Section V.

Model and method
================

In Ref. [@Hiyama97], the present authors already studied $_\Lambda^{ }$$_\Lambda^6$He and $_\Lambda^{ }$$_\Lambda^{\!\!\!10}$Be with the use of $\alpha+\Lambda+\Lambda$ three-body 