  We show that if a split link is obtained from a split link $L$ in $S^3$ by
$1/n$-Dehn surgery along a trivial knot $C$, then the link $L\cup C$ is
splittable. That is to say, it is impossible to obtain a split link from a
split link via a non-trivial twisting. As its corollary, we completely
determine when a trivial link is obtained from a trivial link via a twisting.
