A more thorough explanation:

A loyalty oath is a pledge or promise of allegiance to a particular group, organization, or country. It is a formal declaration of loyalty that is often required of employees, members of the military, and public officials.

An example of a loyalty oath is the Oath of Allegiance that new citizens of the United States must take. This oath requires them to renounce their allegiance to any other country and pledge their loyalty to the United States.

Another example is the loyalty oath that was required of employees during the Cold War era. Many government agencies and universities required employees to sign a loyalty oath to prove that they were not affiliated with communist organizations.

These examples illustrate how a loyalty oath is a formal declaration of loyalty that is often required in certain situations.