can you please revise. thank you

Indeed, the Supreme Court of the United States and the lower federal courts have established the set of rules defining the law and practice of the United States. In particular, regarding the exercise of criminal and civil jurisdiction,in which the United States government sustains jurisdiction over foreign vessels and persons on board of them in territorial water.
For example, in the Wildenhus’s case in which a Belgian native was accused of homicide against another Belgian on board a Belgian vessel anchored in the port of Jersey City (1498). Here, the Supreme Court of the United States sustained the jurisdiction of the lower courts of New Jersey based on the territory sovereign principle under customary law. Although, treaty in place, between the Belgian government and the United States, in which under Article 11 (21 Stat 776) of the treaty provided the Belgian government consuls with “exclusive charge of internal order of the merchant vessels of their nation” and that the State of New Jersey is without jurisdiction. The circuit court refused to deliver the prisoner to the consul ( Smit1498).
However, the United States under the same treaty invoked sections 751 and 753 of the Revised Statutes, in which it gives the power to the United States to issue writ of habeas corpus to determine whether the Belgian man who is in the custody of the U. S. authorities charged with the commission of a crime within the port against the laws of the state, is excused from local jurisdiction under the provisions of a treaty between the United States and Belgium which the vessel belongs (Smit1498).

Indeed, the Supreme Court of the United States and the lower federal courts have established a set of rules that define the law and practice in the United States. These rules govern the exercise of criminal and civil jurisdiction, including cases involving foreign vessels and individuals on board those vessels in territorial waters.

For example, in the Wildenhus's case, a Belgian native was accused of homicide against another Belgian on board a Belgian vessel anchored in the port of Jersey City in 1498. The Supreme Court of the United States upheld the jurisdiction of the lower courts in New Jersey based on the principle of territorial sovereignty under customary law.

It is important to note that there was a treaty in place between the Belgian government and the United States. Under Article 11 of this treaty, the Belgian government's consuls were given exclusive authority over the internal order of their nation's merchant vessels. Based on this provision, the State of New Jersey argued that it did not have jurisdiction in this case. However, the circuit court ruled against the State of New Jersey and refused to hand over the prisoner to the consul.

In response, the United States invoked sections 751 and 753 of the Revised Statutes, which grant the United States the power to issue a writ of habeas corpus. This writ is used to determine whether the Belgian man, who is in the custody of U.S. authorities and charged with committing a crime within the port, is exempt from local jurisdiction due to the provisions of the treaty between the United States and Belgium, to which the vessel belongs.

In summary, the case of Wildenhus's exemplifies the interplay between territorial sovereignty, customary law, and the provisions of international treaties in determining jurisdiction in cases involving foreign vessels and individuals on board those vessels in U.S. territorial waters.