The Word 'Israel' is mentioned 2300 times in the Bible. In the Hebrew and Chaldee Dictionary #3478 Yisrael from #8260 and #410 he will rule as God and typical of his posterity - Israel.
"And he said, Thy name shall be called no more Jacob, but Israel: for as a prince hast thou power with God and with men, and hast prevailed." (Genesis 32:28)

"Yet the number of the children of Israel shall be as the sand of the sea, which cannot be measured nor numbered; and it shall come to pass, that in the place where it was said unto them, Ye are not my people, there it shall be said unto them, Ye are the sons of the living God". (Hosea 1:10)

"And the remnant of Jacob shall be among the Gentiles in the midst of many people as a lion among the beasts of the forest, as a young lion among the flocks of sheep: who, if he go through, both treadeth down, and teareth in pieces, and none can deliver." (Micah 5:8)

"Who is a God like unto thee, that pardoneth iniquity, and passeth by the transgression of the remnant of his heritage? he retaineth not his anger for ever, because he delighteth in mercy " (.Micah 7:18)

"Esaias also crieth concerning Israel, Though the number of the children of Israel be as the sand of the sea, a remnant shall be saved:" (Romans 9:27)

"Even so then at this present time also there is a remnant according to the election of grace." (Romans 11:5)

"And the dragon was wroth with the woman (Israel), and went to make war with the remnant of her seed, which keep the commandments of God, and have the testimony of Jesus Christ. (Messianic Israel)" (Revelation 12:17)

"As concerning the gospel, they are enemies for your (Gentiles) sakes: but as touching the election, they are beloved for the fathers' sakes." (Romans 11:28)

"A new heart also will I give you (Remnant of Israel), and a new spirit will I put within you: and I will take away the stony heart out of your flesh, and I will give you an heart of flesh." (Ezekiel 36:26)

"For the gifts and calling of God are without repentance. For as ye in times past have not believed God, yet have now obtained mercy through their unbelief: Even so have these also now not believed, that through your mercy they also may obtain mercy. For God hath concluded them all in unbelief, that he might have mercy upon all." (Romans 11:29-32)

"The remnant of Israel shall not do iniquity, nor speak lies; neither shall a deceitful tongue be found in their mouth: for they shall feed and lie down, and none shall make them afraid. Sing, O daughter of Zion; shout, O Israel; be glad and rejoice with all the heart, O daughter of Jerusalem. The LORD hath taken away thy judgments, he hath cast out thine enemy: the king of Israel, even the LORD, is in the midst of thee: thou shalt not see evil any more. In that day it shall be said to Jerusalem, Fear thou not: and to Zion, Let not thine hands be slack. The LORD thy God in the midst of thee is mighty; he will save, he will rejoice over thee with joy; he will rest in his love, he will joy over thee with singing. I will gather them that are sorrowful for the solemn assembly, who are of thee, to whom the reproach of it was a burden. Behold, at that time I will undo all that afflict thee: and I will save her that halteth, and gather her that was driven out; and I will get them praise and fame in every land where they have been put to shame. At that time will I bring you again, even in the time that I gather you: for I will make you a name and a praise among all people of the earth, when I turn back your captivity before your eyes, saith the LORD." (Zephaniah 3:13-20)

"Who are Israelites; to whom pertaineth the adoption, and the glory, and the covenants, and the giving of the law, and the service of God, and the promises; Whose are the fathers, and of whom as concerning the flesh Christ came, who is over all, God blessed for ever. Amen. Not as though the word of God hath taken none effect. For they are not all Israel, which are of Israel:" (Romans 9:4-6)

Adoption in Lexical Aids to the New Testament #5206 Hulothesia: from (5207) son and (5087) to place. Adoption, receiving another into the relationship of a child of someone. Used by Paul in Romans 9:4 with reference to the filial relationship into which Israel was admitted by election to God.
In other words Gentile believers who repented of their sins and accepted Jesus Christ as their Lord and Savior, because they received the Holy Spirit, have been permitted as adopted sons and daughters of God into the relationship between God and the Sons of God elect Israel. To rightly explain that relationship the Apostle Paul likened Israel and the Gentile believers to a natural olive tree (Israel) and a wild olive tree (Gentile Believers).

"For if the firstfruit be holy, the lump is also holy: and if the root be holy, so are the branches. And if some of the branches be broken off, and thou, being a wild olive tree, wert graffed in among them, and with them partakest of the root and fatness of the olive tree; Boast not against the branches. But if thou boast, thou bearest not the root, but the root thee. Thou wilt say then, The branches were broken off, that I might be graffed in. (Romans 11:16-19)

"For in Christ Jesus neither circumcision (Israel) availeth any thing, nor uncircumcision (Gentiles), but a new creature. And as many as walk according to this rule, peace be on them, and mercy, and upon the Israel of God (The Body of Christ Jesus, His Church, His Bride, His Sons of God). " (Galatians 6:15-16)

"And so all Israel shall be saved: as it is written, There shall come out of Sion the Deliverer, and shall turn away ungodliness from Jacob:" (Romans 11:26)

"For God hath concluded them all in unbelief, that he might have mercy upon all. O the depth of the riches both of the wisdom and knowledge of God! how unsearchable are his judgments, and his ways past finding out! For who hath known the mind of the Lord? or who hath been his counsellor? (Romans 11:32-34)

Praise God forever more, Amen