The goal of my answer is only to provide recent references.
I warmly recommend these two bits of T. Y Lam's book [1]:

- §I.8, for examples where transvections fail to generate $SL_n(R)$  
- the second to last paragraph of §VIII.12 for other interesting examples of rings $R$ satisfying $SL_2(R) = E_2(R)$ or its negative.

And also B. Magurn's latest article on generalized Euclidean group rings [2].

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[1] Serre's problem on projective modules, T. Y. Lam, 2006  
[2] On a note from Oliver concerning generalized Euclidean group rings. B. Magurn, 2014