Analytic factorization of Lie group representations

For every moderate growth representation of a real Lie group G on a Frechet space E, we prove a factorization theorem of Dixmier--Malliavin type for the space of analytic vectors E^{\omega}. There exists a natural algebra of superexponentially decreasing analytic functions A(G), such that E^{\omega} = A(G) * E^{\omega}. As a corollary we obtain that E^\omega coincides with the space of analytic vectors for the Laplace--Beltrami operator on G.