  For an arbitrary ring $A$, we study the abelianization of the elementary
group $\textrm{E}_2(A)$. In particular, we show that for a commutative ring $A$
there exists an exact sequence \[ K_2(2,A)/C(2,A) \to A/M \to
\textrm{E}_2(A)^\textrm{ab} \to 1, \] where $C(2,A)$ is the subgroup of the
Steinberg group $\textrm{St}(2,A)$ generated by the Steinberg symbols and $M$
is the additive subgroup of $A$ generated by $x(a^2-1)$ and $3(b+1)(c+1)$, with
$x\in A$, $a,b,c \in A^{\times}$.
