Satellite fully positive braid links are braided satellites of fully positive braid links

A link in [Formula: see text] is a fully positive braid link if it is the closure of a positive braid that contains at least one full twist. We show that a fully positive braid link is a satellite link if and only if it is the satellite of a fully positive braid link [Formula: see text] such that the pattern is a positive braid that contains sufficiently many full twists, where the number of necessary full twists only depends on [Formula: see text]. The proof uses various techniques and aspects of braids. As an application, we give a characterization of the unknot by the property that a certain braided satellite is a (fully) positive braid knot.