This is related to [Symplectic group over $\mathbb{Z}/p\mathbb{Z}$ is generated by its root subgroups](https://mathoverflow.net/questions/401996/symplectic-group-over-mathbbz-p-mathbbz-is-generated-by-its-root-subgroup). There I was told that in general, the symplectic group $\text{Sp}\_{2n}(R)$ is not generated by its root subgroups and a maximal torus.


My question now is: for which commutative rings $R$ is $\text{Sp}\_{2n}(R)$ generated by its root subgroups and a maximal torus?



When is the symplectic group over a commutative ring generated by its root subgroups and a maximal torus?
Since $\operatorname{Sp}\_{2\ell}$ is simply-connected, there is no need for the maximal torus. So the question is about the triviality of $\operatorname{K\_1}(\mathsf{C}\_\ell, R) = \operatorname{Sp}(2\ell,R)/\operatorname{Ep}(2\ell,R)$, where $\operatorname{Ep}(2\ell,R)$ is the elementrary symplectic group, that is, the subgroup of $\operatorname{Sp}(2\ell,R)$ generated by its root subgroups.


A lot of cases where $\operatorname{K\_1}(\mathsf{C}\_\ell,R)$ is trivial are not specific fot the symplectic case and hold for all Chevalley groups. Below is a(n incomplete) list. I only consider commutative rings.


* $R$ is a ring of stable rank 1 (this includes local and semi-local rings, boolean rings, the ring of algebraic integers, the ring of entire functions, the disc algebra). The proof is a combination of surjective stabiliy for $\operatorname{K}\_1$ (a [paper](https://doi.org/10.4099/math1924.4.77) by M. Stein) and an easy argument for $\operatorname{SL}(2,R)$ (every its element is a product of 4 elementary matrices, this was apparently first noted by H. Bass);
* $R$ is a Dedekind ring of arithmetic type in a number field which is not totally imaginary (this is Theorem 3.6 from [Bass—Milnor—Serre](http://www.numdam.org/articles/PMIHES_1967__33__59_0/));
* $R$ is Euclidean (the same as the stable rank 1 case, but with the Euclidean algorithm for $\operatorname{SL}(2,R)$ and no bound on the number of elementary factors);
* More generally, if $R$ is a ring such that $\operatorname{K}\_1({}\cdot{},R)=1$, it is sometimes possible to show that $\operatorname{K}\_1({}\cdot{},R[x\_1,\ldots,x\_n])=1$, see my answer [here](https://mathoverflow.net/questions/156563/suslins-stability-theorem-for-chevalley-groups/);
* There are many papers dealing with the rings of geometric or analytical origin, for example, in [this paper](https://doi.org/10.1090/proc/14891) by B. Ivarsson, F. Kutzschebauch and E. Løw the authors consider commutative Banach algebras and some rings of continuous functions (they prove that null-homotopic matrices are elementary, but the paper also contains some references for other rings).


