What oath did lincoln claim in his inaugural address?

The answer is: Lincoln claimed to defend the Union and attempted to convince the South that the secession was illegal and impossible.

Lincoln gained massive popularity on the northern states because he advocated for the establishment of slavery, and the south was threatening secession because they relied heavily on slavery as the major workforce that sustain their economy.

Lincoln realized that if the southern states manage to secede from united states, The republican party would receive the largest amount of blame for the separation, and their chance to free the slaves in southern states were gone. Because of this, he swore an oath to defend the union and prevent the south from seceding in his inaugural address.