The goal of my answer is only to provide recent references. I warmly recommend this 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].
[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