Hypernuclei are a convenient laboratory to study the baryon-baryon weak interaction and associated effective weak Hamiltonian. The strangeness changing process, in which a L hyperon in nuclear matter converts to a neutron with a release of up to 176 MeV, provides a clear signal for a conversion of an s quark to a d quark. It is shown how the nuclear structure aspects of the problem, often an unwelcome detail of calculations attempting to understand a basic two-body LN NN interaction, can be used to pick out components of the effective weak Hamiltonian. It is well known that removing one neutron from ⁹Be results in ⁸Be^* with a subsequent aa decay. Through this process it would be possible to identify final states of the residual nucleus. So, due to this salient feature of the core nuclei ⁹Be and ⁹B, it may be possible to measure the branching fractions G^n(p)_aai for the exclusive decays of the _L¹⁰ Be and _L¹⁰B hypernuclei. These branching fractions G^n(p)_aai depend on various combinations of four matrix elements, hence their study offers a unique possibility of determining all needed matrix elements of the weak interaction and in such a way localizing the difficulties of the hypernuclear nonmesonic decay puzzle. Recently the Nuclotron accelerator at JINR (Dubna) has supplied the first extracted beam of medium-energy ions. This opens new opportunities of performing hypernuclear experiments. With the new trigger tuned to search for two a-particles, the branching fractions Gⁿ_aai (_L¹⁰ Be) and G^p_aai (_L¹⁰ B) will be measured.

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