Steinberg group over a unital ring

Definition
Suppose $$R$$ is a (associative) unital ring and $$n$$ is a natural number. The Steinberg group of degree $$n$$ over $$R$$ (also called the unstable Steinberg group), denoted $$\operatorname{St}_n(R)$$ or $$\operatorname{St}(n,R)$$, is defined by the following presentation:


 * The generating set is as follows: For every element $$\lambda \in R$$ and for $$1 \le i,j \le n$$, $$i \ne j$$, we have a generator $$e_{ij}(\lambda)$$.
 * The relations are as follows. In all cases, $$\lambda,\mu$$ vary freely over all of $$R$$, and are allowed to be equal or distinct.

Case $$n= 1 $$
The case $$n = 1$$ gives a trivial group because there are no generators and no relations. This is not of interest.

Case $$n = 2$$
The case $$n = 2$$ is somewhat different from the case $$n \ge 3$$. For $$n = 2$$, we simply get a free product of two copies of the additive group of $$R$$. This is because there are no relations of the commutator type, and hence, there are no relations connecting the $$e_{12}(\lambda)$$ with the $$e_{21}(\lambda)$$ type generators.

There is an alternative definition of Steinberg group some people use for $$n = 2$$ that is intended to remedy this problem. What is it?

Stable version
The stable Steinberg group over a unital ring is similar to the above except that we have no size restrictions on $$i$$ and $$j$$.

Facts
For every $$R$$ and $$n$$, there is a standard homomorphism from the Steinberg group to the group generated by elementary matrices over a unital ring $$E_n(R)$$. This homomorphism sends the generator $$e_{ij}(\lambda)$$ to the elementary matrix $$e_{ij}(\lambda)$$, i.e., the matrix with $$1$$s on the diagonal, $$\lambda$$ in the $$(ij)^{th}$$ entry, and $$0$$s elsewhere. When $$R$$ is a field, the group $$E_n(R)$$ coincides with the special linear group $$SL_n(R)$$ (see Elementary matrices of the first kind generate the special linear group over a field).

Note that $$E_n(R)$$ coinciding with $$SL_n(R)$$ also holds when $$R$$ is a Euclidean domain.