The Scofield Bible Commentary, by Cyrus Ingerson Scofield, , at sacred-texts.com
"My people" is an expression used in the Old Testament exclusively of Israel the nation. It is never used of the patriarchs, Abraham, Isaac, and Jacob. (Mat 2:6).
"Israel" in Hosea means the ten tribes forming the northern kingdom as distinguished from "Judah" (the tribes of Judah and Benjamin) forming the southern kingdom which adhered to the Davidic family.
(See (Kg1 12:1-21). The promise of verse 10 (Hos 1:10) awaits fulfilment.
See "Israel"; (Gen 12:2); (Gen 12:3); (Rom 11:26).