Smooth representations of a direct product of groups

Question

Let $G_1, G_2$ be Lie groups, and let $E_i$, $i=1,2$ be a smooth representation of $G_i$ in a locally convex complete Hausdorff TVS. Then $E_1\hat\otimes E_2$ is a smooth representation of $G_1\times G_2$ [Warner I, prop. 4.4.1.10], where $\hat\otimes$ denotes the projective tensor product.

Is it true that any irreducible smooth representation of $G_1\times G_2$ is of the form $E_1\hat\otimes E_2$ for irreducible smooth $E_i$? If not, is it true under some additional assumptions on $G_i$?