  On the dual space of \textit{extended structure}, equations governing the
collective motion of two mutually interacting Lie-Poisson systems are derived.
By including a twisted 2-cocycle term, this novel construction is providing the
most general realization of (de)coupling of Lie-Poisson systems. A double cross
sum (matched pair) of 2-cocycle extensions are constructed. The conditions are
determined for this double cross sum to be a 2-cocycle extension by itself. On
the dual spaces, Lie-Poisson equations are computed. We complement the
discussion by presenting a double cross sum of some symmetric brackets, such as
double bracket, Cartan-Killing bracket, Casimir dissipation bracket, and
Hamilton dissipation bracket. Accordingly, the collective motion of two
mutually interacting irreversible dynamics, as well as mutually interacting
metriplectic flows, are obtained. The theoretical results are illustrated in
three examples. As an infinite-dimensional physical model, decompositions of
the BBGKY hierarchy are presented. As finite-dimensional examples, the coupling
of two Heisenberg algebras and coupling of two copies of $3D$ dynamics are
studied.
