"The oath of the Lord shall be between the two of them" — the Mekhilta focuses on the divine Name used in this verse. The oath is described as "the oath of the Lord" — using the Tetragrammaton, God's four-letter Name. This is not a generic oath. It is an oath invoking the most sacred Name in Judaism.
From this specification, the Mekhilta derives a universal rule about oaths. Every oath mentioned in the Torah — even those that do not explicitly specify which Name is used — must be sworn using the Tetragrammaton. Since all oaths in the Torah are stated without specifying the formula, and one oath is qualified as being "by the Lord" (using the four-letter Name), all oaths inherit that qualification.
This principle means that every judicial oath in Jewish law carried the weight of God's most sacred Name. When a guardian swore that he did not mishandle a deposit, or when a litigant swore to the truth of his claim, they were not making a casual promise. They were invoking the Ineffable Name itself. The gravity of this act was meant to ensure honesty. A person willing to lie might hesitate before attaching God's holiest Name to a false statement.
The Mekhilta treated the single qualified oath in this verse as a template that defined the standard for all judicial oaths in the Torah. One explicit specification set the rule for every case that left the formula unstated.