If the second inauguration of Mayor Bill de Blasio on Monday lacked some of the excitement of his first go-round, the undercard at the event — a pair of speeches by the newly re-elected comptroller, Scott M. Stringer, and the public advocate, Letitia A. James — provided at least a hint of dramatic tension.

Both politicians are in the second and final terms of their offices and are already being talked about in political circles as potential rivals in the Democratic primary for mayor in 2021. And so, for each, the inauguration address was the unofficial opening stump speech of what is likely to be a protracted, if not yet announced, campaign to succeed Mr. de Blasio, a Democrat.

Mr. Stringer, in his speech, made a nod to consensus, calling the mayor and Ms. James his “partners in government,” while pointedly focusing on the economic inequality and poverty that still bedevil New York City four years after Mr. de Blasio promised to end what he termed the “tale of two cities.”

“More New Yorkers live in poverty than there are people in Philadelphia or Phoenix,” Mr. Stringer said in an address that included no direct mention of the job of comptroller. “If we become a place where the entrance fee is a $2 million condo and where only the most privileged can prosper, then we will betray the promise of New York.”