According to Scripture, then as we have already noted, that Jesus Christ Himself is the True Israel:
Hosea 11:1 - When Israel [was] a child, then I loved him, and called my son out of Egypt.
This passage is directly cited by the Gospel of Matthew and clearly stated that it was "fulfilled" in Christ Jesus:
Matthew 2:13 - And when they were departed, behold, the angel of the Lord appeareth to Joseph in a dream, saying, Arise, and take the young child and his mother, and flee into Egypt, and be thou there until I bring thee word: for Herod will seek the young child to destroy him.
Matthew 2:14 - When he arose, he took the young child and his mother by night, and departed into Egypt:
Matthew 2:15 - And was there until the death of Herod: that it might be fulfilled which was spoken of the Lord by the prophet, saying, Out of Egypt have I called my son.
The name "Israel" is a Spiritual name, not a Physical one. Consider the origin of it to begin with:
Genesis 32:28 - 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.
This is given twice, for a witness:
Genesis 35:10 - And God said unto him, Thy name [is] Jacob: thy name shall not be called any more Jacob, but Israel shall be thy name: and he called his name Israel.
Yet, notice, that since Christ Jesus is "Israel", who then are His "children" (Exodus 31:13; see also Isaiah 56:1-8 (carefully) along with John 10:16, etc)? Scripture declares it:
Hebrews 2:10 - For it became him, for whom [are] all things, and by whom [are] all things, in bringing many sons unto glory, to make the captain of their salvation perfect through sufferings.
Hebrews 2:11 - For both he that sanctifieth and they who are sanctified [are] all of one: for which cause he is not ashamed to call them brethren,
Notice again from the New Testament in the same place, and from where it draws from in the Old Testament:
Hebrews 2:13 - And again, I will put my trust in him. And again, Behold I and the children which God hath given me.
This is directly citing Isaiah 8, please notice carefully:
Isaiah 8:10 - Take counsel together, and it shall come to nought; speak the word, and it shall not stand: for God [is] with us.
Isaiah 8:11 - For the LORD spake thus to me with a strong hand, and instructed me that I should not walk in the way of this people, saying,
Isaiah 8:12 - Say ye not, A confederacy, to all [them to] whom this people shall say, A confederacy; neither fear ye their fear, nor be afraid.
Isaiah 8:13 - Sanctify the LORD of hosts himself; and [let] him [be] your fear, and [let] him [be] your dread.
Isaiah 8:14 - And he shall be for a sanctuary; but for a stone of stumbling and for a rock of offence to both the houses of Israel, for a gin and for a snare to the inhabitants of Jerusalem.
Isaiah 8:15 - And many among them shall stumble, and fall, and be broken, and be snared, and be taken.
Isaiah 8:16 - Bind up the testimony, seal the law among my disciples.
Isaiah 8:17 - And I will wait upon the LORD, that hideth his face from the house of Jacob, and I will look for him.
Isaiah 8:18 - Behold, I and the children whom the LORD hath given me [are] for signs and for wonders in Israel from the LORD of hosts, which dwelleth in mount Zion.
Notice, that the entire context is speaking about Jesus Christ, the "God with us", and his ["my"] "disciples", even the "children whom the LORD [Father] hath given me [Jesus]", and that the "testimony" and the "Law" of God, the Ten Commandments, were to be "sealed" [by the Holy Spirit] "among" them.
See John 17, but also:
John 1:12 - But as many as received him, to them gave he power to become the sons of God, [even] to them that believe on his name:
and, notice the fulfillment of the "my disciples" of Isaiah 8:
Matthew 26:18 - And he said, Go into the city to such a man, and say unto him, The Master saith, My time is at hand; I will keep the passover at thy house with my disciples.
Mark 14:14 - And wheresoever he shall go in, say ye to the goodman of the house, The Master saith, Where is the guestchamber, where I shall eat the passover with my disciples?
Luke 22:11 - And ye shall say unto the goodman of the house, The Master saith unto thee, Where is the guestchamber, where I shall eat the passover with my disciples?
John 8:31 - Then said Jesus to those Jews which believed on him, If ye continue in my word, [then] are ye my disciples indeed;
John 13:35 - By this shall all [men] know that ye are my disciples, if ye have love one to another.
John 15:8 - Herein is my Father glorified, that ye bear much fruit; so shall ye be my disciples.
John 13:33 - Little children, yet a little while I am with you. Ye shall seek me: and as I said unto the Jews, Whither I go, ye cannot come; so now I say to you.
All of the Promises of God, are only "Yea" and "Amen" in Christ Jesus, and in no other:
2 Corinthians 1:20 - For all the promises of God in him [are] yea, and in him Amen, unto the glory of God by us.
The True Olive Tree of Romans 11, is Christ Jesus Himself, and this may be seen from many further texts.
We can then see what a True Israelite is when we begin to look a little further.
The physical Jew, has no hope, except in coming to, accepting and believing on Christ Jesus, for it is written of them:
Matthew 21:43 - Therefore say I unto you, The kingdom of God shall be taken from you, and given to a nation bringing forth the fruits thereof.
Matthew 23:38 - Behold, your house is left unto you desolate.
Luke 13:35 - Behold, your house is left unto you desolate: and verily I say unto you, Ye shall not see me, until [the time] come when ye shall say, Blessed [is] he that cometh in the name of the Lord.
Matthew 3:10 - And now also the axe is laid unto the root of the trees: therefore every tree which bringeth not forth good fruit is hewn down, and cast into the fire.
Luke 3:9 - And now also the axe is laid unto the root of the trees: every tree therefore which bringeth not forth good fruit is hewn down, and cast into the fire.
Mark 11:14 - And Jesus answered and said unto it, No man eat fruit of thee hereafter for ever. And his disciples heard [it].
Mark 11:20 - And in the morning, as they passed by, they saw the fig tree dried up from the roots.
Romans 9:6 - Not as though the word of God hath taken none effect. For they [are] not all Israel, which are of Israel:
The "Israel of God", is not the "Israel after the flesh":
1 Corinthians 10:18 - Behold Israel after the flesh: are not they which eat of the sacrifices partakers of the altar?
Galatians 6:16 - And as many as walk according to this rule, peace [be] on them, and mercy, and upon the Israel of God.
For one to be an "Israelite indeed", they must accept Jesus Christ:
John 1:47 - Jesus saw Nathanael coming to him, and saith of him, Behold an Israelite indeed, in whom is no guile!
John 1:48 - Nathanael saith unto him, Whence knowest thou me? Jesus answered and said unto him, Before that Philip called thee, when thou wast under the fig tree, I saw thee.
John 1:49 - Nathanael answered and saith unto him, Rabbi, thou art the Son of God; thou art the King of Israel.