  Let $K$ be a number field and ${\mathcal O}$ be the ring of $S$-integers in
$K$. Morgan, Rapinchuck, and Sury have proved that if the group of units
${\mathcal O}^{\times}$ is infinite, then every matrix in ${\rm SL}_2({\mathcal
O})$ is a product of at most $9$ elementary matrices. We prove that under the
additional hypothesis that $K$ has at least one real embedding or $S$ contains
a finite place we can get a product of at most $8$ elementary matrices. If we
assume a suitable Generalized Riemann Hypothesis, then every matrix in ${\rm
SL}_2({\mathcal O})$ is the product of at most $5$ elementary matrices if $K$
has at least one real embedding, the product of at most $6$ elementary matrices
if $S$ contains a finite place, and the product of at most $7$ elementary
matrices in general.
