Let $G, H$ be Lie groups and $A < G, B < H$ Lie subgroups. Is it true that the smooth manifolds $(G \times H)/(A \times B)$ and $(G/A) \times (H/B)$ are canonically diffeomorphic?
I am trying, relying on (given, for example, in Alexander Kirillov's textbook) an explicit assignment of a smooth structure on adjacency classes, to show this. It seems to me that the trivial reasoning with local charts works, but my doubts remain.