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

In this article we study the first, the second and the third homology groups of the elementary group E₂(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∈ (A×)² or |A×/(A×)²|≤ 4.