This answer provides references for the facts mentioned in the comments. More on this topic can be found in [1, Section 9].
For $R$ a unital ring, we denote by $\text{E}_n(R)$ the subgroup of $\text{GL}_n(R)$ generated by all the transvections $$e_{ij}(r) = I_n + r \epsilon_{ij}$$ (a.k.a the elementary matrices over $R$) with $r \in R, 1\le i \neq j \le n$ and where $I_n$ is identity matrix and $\epsilon_{ij}$ is the matrix whose $(i,j)$-entry is $1$ while all its other entries are zero.
Claim. Let $R$ be a unital ring of finite stable rank $\text{sr(R)}$. Then we have $$[\text{GL}_n(R), \text{GL}_n(R)] = \text{E}_n(R)$$ for every $n > \text{sr}(R)$.
Proof. This follows for instance from Vaserstein's Injective Stability Theorem [1, Theorem 10.15].
Corollary. Let $R$ be a commutative and unital ring of Krull dimension at most $1$, e.g., a principal ideal domain. Then $[\text{GL}_n(R), \text{GL}_n(R)] = \text{E}_n(R)$ for every $n > 2$.
Proof. As the stable rank of a commutative and unital ring of Krull dimension at most $d$, is at most $d + 1$ [2, Corollary 2.3], the result follows from the previous claim.
For $n = 2$, Section 9 of P. M. Cohn's [3] contains a wealth of valuable results. For instance it follows from [3, Proposition 9.2] that $$[\text{GL}_2(R), \text{GL}_2(R)] = \text{E}_2(R)$$ if $R$ is Euclidean and $1$ is the sum of two units of $R$ (e.g., $R = \mathbb{Z}[\frac{1}{2n}]$ where $n \in \mathbb{N}_{> 0}$ or $R = k[X]$ where $k$ is a field with at least $3$ elements). See also , among others, Cohn's generalization of the result of Hua and Reiner [1, Theorem 9.4].
- [1] P. M. Cohn, "On the structure of the $GL_2$ of a ring", 1966.
- [2] R. Heitmann, "Generating non-Noetherian modules efficiently", 1984.
- [3] B. Magurn, "An algebraic introduction to $K$-theory", 2002.