Proposition 1

Every compact topological group G is isomorphic to a closed subgroup of a product of compact Lie groups.

Denote by $\widehat{G}$ the set of classes of irreducible continuous unitary representations of G on finite dimensional complex Hilbert spaces (Spectral Theory, in preparation). For all $u\in \widehat{G}$, let $Hu$ be the space of $u$ and $\rho_u: G\rightarrow \mathbf{U}(H_u)$ the homomorphism associated to $u$. By the Peter-Weyl theorem (Spectral Theory, in preparation), the continuous homomorphism $\rho = (\rho_u)_{u\in\widehat{G}}$ from G to $\prod_{u\in\widehat{G}}\mathbf{U}(H_u)$ is injective; since G is compact, $\rho$ induces an isomorphism

from G onto a closed subgroup of the group $\prod_{u\in\widehat{G}}\mathbf{U}(H_u)$.