The promise to leave an inheritance for the Davidic dynasty after the division of the kingdom stems from God's enduring commitment to His choice of David and the city of Jerusalem. As the nation splits, ten tribes are handed over to Jeroboam, leaving a remaining portion specifically for Solomon [ביאור שטיינזלץ].
This division creates an apparent mathematical mismatch, as ten tribes given away plus one remaining tribe only accounts for eleven of the twelve tribes of Israel. This is resolved by understanding that the tribes of Judah and Benjamin are counted together as a single unit. They are considered one entity because they were deeply united through their shared land and their joint connection to Jerusalem [רד״ק]. God preserves this unified inheritance for the royal house to honor David, acting as a direct continuation of the divine promise He had given to him in the past [מלבי״ם].