Consider the group $\mathrm{Diff}(\mathbb R^n)$ of smooth diffeomorphisms. It has two interesting subgroups:

- the orthogonal group $O(n)$,
- the group of "diffeomorphisms applied along each axis" $\mathrm{Diff}(\mathbb R)^n$, so that $f(x_1, \dotsc, x_n) = \big(f_1(x_1), \dotsc, f_n(x_n)\big)$, where each $f_i \in \mathrm{Diff}(\mathbb R)$.

**My question is whether the group generated by these two subgroups is the whole** $\mathrm{Diff}(\mathbb R^n)$ **or not.**
(I suspect it is *not* for $n\ge 2$, so any results characterizing this subgroup would be very welcome. I did a literature search and I have not found the answer to this problem, but I am not an expert, so any references on this problem or similar would be great).

- In dimension $n=1$ this is trivially true.
- Sole $\mathrm{Diff}(\mathbb R)^n \neq \mathrm{Diff}(\mathbb R^n)$ for $n\ge 2$, as the elements of the first group can only have diagonal jacobians.
- The linearised version of the problem is true: $GL(n)$ is the same as the group generated by $O(n)$ and invertible diagonal matrices $(\mathbb R^\times)^n$, which is a corollary of singular value decomposition.