What caused one house (Judah) to retain its identity and the other (Israel) to lose it? God gives the answer—the Sabbath. Whedon's Commentary on the Bible (at verse 12) states, “The Sabbath was the visible sign to the Hebrews and to the world that they were his, and that he was theirs.”

A sign identifies. An example is a burqa, the long, loose garment covering the whole body from head to feet, worn in public by many Muslim women. A woman wearing a burqa is an unmistakable sign that she is Allah's and that he is her god. It is a sign that separates and identifies.

With these verses in chapter 20 of Ezekiel, God guarantees that if His people keep the Sabbath—the sign that signifies who they are and who they worship—that sign would assure their identity. To be separate requires a definable identity. God, by their observance of the Sabbath, guaranteed they would remain separate and set apart, preserved as a select people unmixed with the nations.

Throughout history, the house of Judah has continued to observe the Sabbath. As a result, they have retained their identity. On the other hand, the house of Israel rejected the Sabbath, and true to Ezekiel 20, they have disappeared from view. They have lost their identity. They no longer have the sign that tells them, or the rest of the world, who they are.

The house of Israel chose to rebel against the Sabbath, a proclivity that began as far back as the wilderness journey:

Yet the house of Israel rebelled against Me in the wilderness; they did not walk in My statutes; they despised My judgments, 'which, if a man does, he shall live by them'; and they greatly defiled My Sabbaths. Then I said I would pour out My fury on them in the wilderness, to consume them. (Ezekiel 20:13)

So, the Bible clearly and consistently identifies as Israel those nations that have become known as the "Ten Lost Tribes," rather than the house of Judah.