Okla. Stat. tit. 22 § 1058
If an appeal is taken and the appeal bond given as provided in the preceding section, said bond shall be conditioned that the defendant will appear, submit to and perform any judgment rendered by the Criminal Court of Appeals or the court in which the original judgment was rendered in the further progress of the cause, and will not depart without leave of the court. After the determination of the appeal in the Criminal Court of Appeals, or if the appeal is not perfected as provided by law, the defendant may be surrendered by the sureties to the proper authorities for the execution of the sentence. If the defendant be adjudged to be incarcerated in any penal institution and/or to pay a fine, said sureties shall be relieved of liability for such fine and costs upon surrender of the defendant to the proper authorities for incarceration pursuant to the judgment and prior to forfeiture of the bond. If no bond be given the appeal shall not stay execution of the judgment, except in capital cases or where otherwise specifically provided by law. If pending the appeal the bond be given, a further execution of the judgment shall be stayed and the defendant released pending the determination of the appeal. In all cases where the sentence is for a crime not bailable the defendant shall be confined in the penitentiary pending the appeal.
Okla. Stat. tit. 22, § 1058