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In the United States, the term "treaty" has a different, more restricted legal sense than exists in international law. United States law distinguishes what it calls treaties from executive agreement, congressional-executive agreements, and sole executive agreements. All four classes are equally treaties under international law; they are distinct only from the perspective of internal American law. The distinctions are primarily concerning their method of approval. Whereas treaties require advice and consent by two-thirds of the Senators present, sole executive agreements may be executed by the President acting alone. Some treaties grant the President the authority to fill in the gaps with executive agreements, rather than additional treaties or protocols. And finally, congressional-executive agreements require majority approval by both the House and the Senate, either before or after the treaty is signed by the President.

Under US law, what primarily distinguishes treaties, executive agreements, congressional-executive agreements, and sole executive agreements?
their method of approval