Matthew 10:5-7 KJVS
 These twelve Jesus sent forth, and commanded them, saying, Go not into the way of the Gentiles, and into any city of the Samaritans enter ye not:  But go rather to the lost sheep of the house of Israel.  And as ye go, preach, saying, The kingdom of heaven is at hand.
Much has been written about the identity of the lost sheep of the house of Israel. Usually a history of the two kingdoms of the Old Testament houses of Judah and Israel is given. There has been much speculation and different theories about who the lost tribes of Israel are when the Lord walked this earth and in the present day. Many miss the obvious clues in the New Testament that point to who the lost sheep of the house of Israel are. First one should examine the above words of our Lord from the gospel of Matthew. He limits the area the twelve disciples can preach in, to the present country of Israel. So obviously the lost sheep of the house of Israel are in the country of Israel while our Lord walked the land. At first glance this makes no sense since the lost sheep of the house of Israel in the Old Testament were carried away to Assyria, heard of no more, and apparently were assimilated in and around that region. But then our Lord would know that and he would not make so obvious a mistake regarding the House of Israel. So there has to be another explanation. Apostle Paul points out there are two Israel. Israel of the flesh and Israel of the promise seed.
Romans 9:6-8 KJVS
 Not as though the word of God hath taken none effect. For they are not all Israel, which are of Israel:  Neither, because they are the seed of Abraham, are they all children: but, In Isaac shall thy seed be called.  That is, They which are the children of the flesh, these are not the children of God: but the children of the promise are counted for the seed.
In short, the Israel of God is Israel of the promise seed. And the promise seed is Jesus Christ and those that are in him, Gal 3:16&29. Apostle Paul make another distinction that takes awhile to wrap your mind around about the law of Moses.
Galatians 4:21-31 KJVS
 Tell me, ye that desire to be under the law, do ye not hear the law?  For it is written, that Abraham had two sons, the one by a bondmaid, the other by a freewoman.  But he who was of the bondwoman was born after the flesh; but he of the freewoman was by promise.  Which things are an allegory: for these are the two covenants; the one from the mount Sinai, which gendereth to bondage, which is Agar.  For this Agar is mount Sinai in Arabia, and answereth to Jerusalem which now is, and is in bondage with her children.  But Jerusalem which is above is free, which is the mother of us all.  For it is written, Rejoice, thou barren that bearest not; break forth and cry, thou that travailest not: for the desolate hath many more children than she which hath an husband.  Now we, brethren, as Isaac was, are the children of promise.  But as then he that was born after the flesh persecuted him that was born after the Spirit, even so it is now.  Nevertheless what saith the scripture? Cast out the bondwoman and her son: for the son of the bondwoman shall not be heir with the son of the freewoman.  So then, brethren, we are not children of the bondwoman, but of the free.
The old covenant (law of Moses) is from the mount Sinai, which gendereth to bondage, which is Aga, and this Agar is mount Sinai in Arabia, and answereth to Jerusalem which now is, and is in bondage with her children. That is a mouthful that makes a important distinction that many miss. The religious system surrounding Herod’s Temple which was based on the law of Moses, children of Agar, are not heirs with the children of the promise seed. The lost sheep of the house of Israel that our Lord sent the twelve disciples to preach saying the kingdom of heaven is at hand, were the Jews that were the children of promise who were to be called out of bondage to the law, Acts 15:10-11. Plus the called out Gentiles that are added to the seed. Apostle Paul makes this clear.
Romans 9:21-26 KJVS
 Hath not the potter power over the clay, of the same lump to make one vessel unto honour, and another unto dishonour?  What if God, willing to shew his wrath, and to make his power known, endured with much longsuffering the vessels of wrath fitted to destruction:  And that he might make known the riches of his glory on the vessels of mercy, which he had afore prepared unto glory,  Even us, whom he hath called, not of the Jews only, but also of the Gentiles?  As he saith also in Osee, I will call them my people, which were not my people; and her beloved, which was not beloved.  And it shall come to pass, that in the place where it was said unto them, Ye are not my people; there shall they be called the children of the living God.
So the commonwealth of Israel Apostle Paul writes about in Ephesians, was comprised initially of Jews who are of the promise seed, and then the Gentiles who are also of the promise seed that were added to the body of Christ.
Ephesians 2:11-19 KJVS
 Wherefore remember, that ye being in time past Gentiles in the flesh, who are called Uncircumcision by that which is called the Circumcision in the flesh made by hands;  That at that time ye were without Christ, being aliens from the commonwealth of Israel, and strangers from the covenants of promise, having no hope, and without God in the world:  But now in Christ Jesus ye who sometimes were far off are made nigh by the blood of Christ.  For he is our peace, who hath made both one, and hath broken down the middle wall of partition between us ;  Having abolished in his flesh the enmity, even the law of commandments contained in ordinances; for to make in himself of twain one new man, so making peace;  And that he might reconcile both unto God in one body by the cross, having slain the enmity thereby:  And came and preached peace to you which were afar off, and to them that were nigh.  For through him we both have access by one Spirit unto the Father.  Now therefore ye are no more strangers and foreigners, but fellowcitizens with the saints, and of the household of God;
I know this is a complicated truth of the gospel. More so since we have been programmed all our lives by movies, books and even Christian teachings that point to Israel of the flesh as God’s chosen people. That the law of Moses was the most important aspect of the Old Testament. Few notice a verse from the Old Testament such as this.
Genesis 21:12 KJVS
 And God said unto Abraham, Let it not be grievous in thy sight because of the lad, and because of thy bondwoman; in all that Sarah hath said unto thee, hearken unto her voice; for in Isaac shall thy seed be called.
This verse has to be understood in light of what Paul wrote in Galatians.
Galatians 3:16-19 KJVS
 Now to Abraham and his seed were the promises made. He saith not, And to seeds, as of many; but as of one, And to thy seed, which is Christ.  And this I say, that the covenant, that was confirmed before of God in Christ, the law, which was four hundred and thirty years after, cannot disannul, that it should make the promise of none effect.  For if the inheritance be of the law, it is no more of promise: but God gave it to Abraham by promise.  Wherefore then serveth the law? It was added because of transgressions, till the seed should come to whom the promise was made; and it was ordained by angels in the hand of a mediator.
Jesus Christ is the promise seed who would be born of the flesh through the genetic linage of Isaac. And the Jews who were in the body of Christ comprises the lost sheep of the tribes of Israel that Jesus sent the twelve to go, preach, saying the kingdom of heaven is at hand, Matt 10:6&7.