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

In this article, we study the first, second, and third homology groups of the elementary group $${\textrm{E}}_2(A)$$ , where A is a commutative ring. In particular, we establish a refined Bloch–Wigner type exact sequence over a semilocal ring (subject to mild restrictions on its residue fields) in which either $$-1 \in ({A^\times })^2$$ or $$|{A^\times }/({A^\times })^2| \le 4$$ .