Does ${\rm Iso}(M\times N)={\rm Iso}(M)\times {\rm Iso}(N)$ hold for product metrics?

Question

I know that ${\rm Iso} (M\times N)={\rm Iso}(M)\times {\rm Iso}(N)$ is not generally correct where $M$ and $N$ are smooth Riemannian manifolds; but

Does ${\rm Iso} (M\times N)={\rm Iso}(M)\times {\rm Iso}(N)$ hold for product metrics?

what about warp-product metrics?

Answer 1

Example: $M=N=\mathbb{R}$ with the standard metric. Then $\operatorname{Isom}\mathbb{E}^2$ is a 3-dimensional Lie group, so obviously cannot be the Cartesian square of $\operatorname{Isom}\mathbb{E}^1$.