If Odin could have heard Loki's thoughts, he might have argued with him. Or he might not have. Asgard, and Odin himself, actually owed a lot to the Trickster. Whether by his own errors, Odin's, or someone else's, Loki had contributed much simply by repairing those errors. That didn't mean Odin wanted him to have all the credit, though. He felt all that had transpired was as it should.
Now he was asking Odin to do something the Valfather might have declined, if he could have. He had no idea what parts Loki's wife and sons still had to play in the grand scheme of it all, but he doubted they would come out on top. It seemed any connected to the Trickster were ill-fated, aside from his two daughters by Glut, at least thus far. However, Loki had played his one trump card, whether he knew it or not.
Their brother-by-blood bond was the strongest tie Odin had to anyone, and he took that oath very seriously. He was already becoming known as the Oath-Breaker, but there were some vows he would not betray under any circumstance. He would swear this promise on Ullr's ring, were it close by.
"I will do what I can, Brother." He had no way of knowing that by the time he got back to Asgard and Hlidskialf to seek out Sigyn's location, it would be too late. Things were unraveling faster than even he knew.
There was little more he could say. The time had come to part ways, possibly forever. "Farewell, Brother," he said softly as he turned to go.