  The existence of cosmological dark matter is in the bedrock of the modern
cosmology. The dark matter is assumed to be nonbaryonic and to consist of new
stable particles. However if composite dark matter contains stable electrically
charged leptons and quarks bound by ordinary Coulomb interaction in elusive
dark atoms, these charged constituents of dark atoms can be the subject of
direct experimental test at the colliders. In such models the excessive
negatively double charged particles are bound with primordial helium in
O-helium atoms, maintaining specific nuclear-interacting form of the dark
matter. The successful development of composite dark matter scenarios appeals
to experimental search for doubly charged constituents of dark atoms, making
experimental search for exotic stable double charged particles experimentum
crucis for dark atoms of composite dark matter. (abridged)
