  The presented splitting lemma extends the techniques of Gromov and
Forstneri\v{c} to glue local sections of a given analytic sheaf, a key step in
the proof of all Oka principles. The novelty on which the proof depends is a
lifting lemma for transition maps of coherent sheaves, which yields a reduction
of the proof to the work of Forstneri\v{c}. As applications we get shortcuts in
the proofs of Forster and Ramspott's Oka principle for admissible pairs and of
the interpolation property of sections of elliptic submersions, an extension of
Gromov's results obtained by Forstneri\v{c} and Prezelj.
