Twenty-second Amendment to the United States Constitution
Some argue that the 22nd Amendment and 12th Amendment bar any two-term president from later serving as vice president as well as from succeeding to the presidency from any point in the presidential line of succession. Others contend that the original intent of the 12th Amendment concerns qualification for "service" (age, residence, and citizenship), while the 22nd Amendment, concerns qualifications for "election", and thus (strictly applying the text) a former two-term president is still eligible to "serve" as vice president (neither amendment restricts the number of times an individual can be elected to the vice presidency), and then succeed to the presidency to serve out the balance of the term (though prohibited from running for election to an additional term).
1796 United States presidential election
With the retirement of Washington after two terms, both parties sought the presidency for the first time. Before the ratification of the 12th Amendment in 1804, each elector was to vote for two persons, but was not able to indicate which vote was for president and which was for vice president. Instead, the recipient of the most electoral votes would become president and the runner-up vice president. As a result, both parties ran multiple candidates for president, in hopes of keeping one of their opponents from being the runner-up. These candidates were the equivalent of modern-day running mates, but under the law they were all candidates for president. Thus, both Adams and Jefferson were technically opposed by several members of their own parties. The plan was for one of the electors to cast a vote for the main party nominee (Adams or Jefferson) and a candidate besides the primary running mate, thus ensuring that the main nominee would have one more vote than his running mate.