Let $\mathcal{M}$ be a smooth finite-dimensional manifold. Let $\text{Diff}(\mathcal{M})$ be its infinite-dimensional Lie group of diffeomorphisms. Let $G_\text{Fin}$ range over all of the finite-dimensional Lie subgroups of $\text{Diff}(\mathcal{M})$. I have an intuition that taking the union of all of these $G_\text{Fin}$ groups should give back $\text{Diff}(\mathcal{M})$. Is this intuition correct?
To rephrase the question negatively, is it possible for there to be an element $d\in\text{Diff}(\mathcal{M})$ which is only a part of infinite-dimensional Lie subgroups of $\text{Diff}(\mathcal{M})$? This would be the case, for instance, there exists some $d_\text{Inf}\in\text{Diff}(\mathcal{M})$ which by itself generates an infinite-dimensional Lie group, $\text{Closure}(d_\text{Inf})=:G_\text{Inf}\subset\text{Diff}(\mathcal{M})$.
If such a $d_\text{Inf}\in G_\text{Inf}\subset\text{Diff}(\mathcal{M})$ exists, then I would be interested to know the size of $G_\text{Inf}$'s orbit on $\mathcal{M}$.
Any thoughts would be much appreciated!