The low dimensional homology groups of the elementary group of degree two

In this article we study the first, the second and the third homology groups of the elementary group ${\text{E}}_2(A)$, where $A$ is a commutative ring. In particular, we prove a refined Bloch-Wigner type exact sequence over a semilocal ring (with some mild restriction on its residue fields) such that $-1\in (A^{\times }{)}^2$ or $\mathrm{\mid }{A}^{\times }\mathrm{/}(A^{\times }{)}^2\mathrm{\mid }\le 4$.