  In the events of peripheral dissociation of relativistic nuclei in the
nuclear track emulsion, it is possible to study the emerging ensembles of He
and H nuclei, including those from decays of the unstable $^8$Be and $^9$B
nuclei, as well as the Hoyle state. These extremely short-lived states are
identified by invariant masses calculated from the opening angles in
2$\alpha$-pairs, 2$\alpha p$- and 3$\alpha$-triplets in the approximation of
conservation of momentum per nucleon of the primary nucleus. In the same
approach, it is possible to search for more complex states. This paper explores
the correlation between the formation of $^8$Be nuclei and the multiplicity of
accompanying $\alpha$-particles in the dissociation of relativistic $^{16}$O,
$^{22}$Ne, $^{28}$Si, and $^{197}$Au nuclei. On this basis, estimates of such a
correlation are presented for the unstable $^9$B nucleus and the Hoyle state.
An enhancement in the $^8$Be contribution to dissociation with the
$\alpha$-particle multiplicity is found. Decays of $^9$B nuclei and Hoyle
states follow the same trend.
