  This paper contains several results about the structure of the congruence
kernel C^(S)(G) of an absolutely almost simple simply connected algebraic group
G over a global field K with respect to a set of places S of K. In particular,
we show that C^(S)(G) is always trivial if S contains a generalized arithmetic
progression. We also give a criterion for the centrality of C^(S)(G) in the
general situation in terms of the existence of commuting lifts of the groups
G(K_v) for v \notin S in the S-arithmetic completion \widehat{G}^(S). This
result enables one to give simple proofs of the centrality in a number of
cases. Finally, we show that if K is a number field and $G$ is K-isotropic then
C^(S)(G) as a normal subgroup of \widehat{G}^(S) is almost generated by a
single element.
