From Pipa’s commentary on Galatians 6:16:
A true believer identifies with the offence of the cross. Paul shows clearly the class of people to whom the blessings belong: those who hold to Christ by faith and identify with the offence of the cross.
Moreover, he clarifies their position by using two titles that define the recipients of God’s blessing. The first is the family title, ‘Brethren.’ Since he has already used this title, I remind you of something that I wrote before. Paul never uses fillers. Every word, every title is chosen with care. Therefore, he is expressing by the term ‘brethren’ (a term that includes men and women) those who belong to God’s family, God’s adopted children, and the co-heirs of the Lord Jesus Christ. So this term is another way of talking about those who follow the rule or walk by the rule. If you are a Christian then you are a brother of Christ; you are a child of God.
Furthermore, because of the issues in this letter about who belongs to the people of God, he calls them, ‘the Israel of God’ (v. 16). This is a common way to address the covenant people, ‘Redeem Israel, O God’ (Ps. 25:22, cf. 130:8). Here he addresses a congregation made up mostly of converted Gentiles, Israel. The Judaizers claimed that these people had to become Jews physically (circumcision) to be part of Israel. Paul has shown that all who believe in Christ are the true seed of Abraham (3:29). Israel is the church. It matters not whether you are circumcised or uncircumcised. If, by faith, you are in union with Christ Jesus, you are God’s Israel.
Hence Paul destroys the idea that God has two people, Israel and the church. No! Here he calls the church Israel. There are not two people. There is the one people of God. We are the children of Abraham; we are God’s Israel. So the blessings of God, the apostolic benediction, are bestowed upon those who are in Christ Jesus.*
*Pipa, J. A., Jr. (2010). Galatians: God’s Proclamation of Liberty. Focus on the Bible Commentary (248). Ross-shire, Scotland: Christian Focus Publications.