Identify the group action underlying centrally extended semidirect product EP-equations

Identify the explicit Lie group and its multiplication (group action) whose Euler–Poincaré variational principle produces the coadjoint evolution system obtained by centrally extending the semidirect product Lie algebra using the diagonal 2‑cocycle \tilde{\sigma}((\xi_1,\xi_2),(\eta_1,\eta_2)) := (\sigma(\xi_1,\eta_1), \sigma(\xi_2,\eta_2)) in Appendix "Centrally extending the semidirect product group"; in particular, construct the group-level action that corresponds to the derived EP-equations where the cocycle contributions are strictly diagonal and do not appear in the coupling terms.

Background

In the appendix subsection "Centrally extending the semidirect product group," the paper defines a central extension of the semidirect product Lie algebra by introducing the diagonal 2-cocycle \tilde{\sigma}((\xi_1,\xi_2),(\eta_1,\eta_2)) := (\sigma(\xi_1,\eta_1), \sigma(\xi_2,\eta_2)). From this construction, the authors derive a coadjoint evolution (Euler–Poincaré) system for variables (m, l, n, o) that displays cocycle contributions strictly on the diagonal blocks.

The authors explicitly note that, although these are EP-equations on a centrally extended semidirect product Lie algebra, the corresponding group-level action (i.e., the multiplication law on the extended group that would yield these EP-equations via reduction) is not identified. Resolving this requires constructing or characterizing the Lie group and its AD/Ad/ad/ad* structures such that the derived algebraic EP-equations are recovered from a genuine group action.