The insurance broker said, Oxygen has agreed to release the executives from their restrictive covenants with immediate effect.
With 21 staff members, the team specialise in sectors such as construction, power and mining, professional indemnity, management risks, binders and facilities.
Gallagher said that clients will experience an uninterrupted service and will continue to be supported by the same specialists who have looked after them at Oxygen and also benefit from the resources and opportunities a larger organisation can provide.
Arthur J Gallagher International CEO David Ross said this agreement represents the culmination of lengthy discussions and builds on the already excellent trading relationship originated by the sale to Gallagher of Oxygen's underwriting business in September 2008.
"The move is a natural extension of this relationship and will complement and enhance Gallagher's existing capabilities," Ross said.
The terms of the transaction are not disclose