Let \(U\) be a unitary representation of a real finite-dimensional Lie group \(G\) on a separable complex Hilbert space \(H\) and let \(dU\) be the corresponding representation of the complex enveloping *-algebra \(L\) of \(G\), defined on a domain \(D\subseteq H\). For \(\Omega\in D\) we have a functional \(S_{(U,\Omega)}(x) = (dU(x)\Omega,\Omega)\) defined for every \(x\in L\). The noncommutative moment problem consists in the following. Let \(S\) be a functional given on \(L\); under what conditions \(S\) may be represented as \(S = S_{(U,\Omega)}?\) The authors consider a very natural class of infinite-dimensional Lie algebras (called AE-algebras) and extend the so-called \(\Phi C^3\)-theory [\textit{M. Flato}, \textit{J. Simon}, \textit{H. Snellman} and \textit{D. Sternheimer}, Ann. Sci. Ec. Norm. Supér., IV. Sér. 5, 423--434 (1972; Zbl 0239.22019), \textit{M. Flato} and \textit{J. Simon}, J. Funct. Anal. 13, 268--276 (1973; Zbl 0257.22012)] in order to get simple sufficient conditions for the solvability of the moment problem in this new setting. The paper is a short summary of results.