I also found this small article helpful;
Under the Law
1. Consider the meaning of the phrase “under the law” in Pauline literature.
Perhaps the best way to understand the phrase is to study the Galatian heresy. Paul used the phrase “under the law” five times in his letter to the Galatians (3:23; 4:4; 4:5; 4:21; 5:18), more than any of his other letters. But how were the false teachers in Galatia attempting to bring God’s people back “under the law?”
First, to be “under the law” is to attempt to secure God’s verdict of justification by the law. Paul says in Galatians 2:21, “I do not nullify the grace of God, for if justification were through the law, then Christ died for no purpose.” The false teachers were teaching Christians that they had to keep the whole law of God for their justification, not only the Ten Commandments, but circumcision and the Jewish food laws and festivals. This was a false gospel (Gal 1:6-7).
Second, to be “under the law” is to attempt to obtain the Spirit through obedience to the law. Paul says in Galatians 3:2, “Let me ask you only this: Did you receive the Spirit by works of the law or by hearing with faith?” Thus, to seek not only the objective blessing of justification by the law, but also the subjective blessing of the Spirit by the law is to be “under the law.” The Judaizers were teaching that to obtain the Spirit of Christ, one must obey the law of God. That is a “bewitching” heresy.
Third, to be “under the law” is to “rely on the works of the law.” Paul says in Galatians 3:10, “For all who rely on works of the law are under a curse; for it is written, ‘Cursed be everyone who does not abide by all things written in the Book of the Law and do them.’” Thus, all who rely on the law, meaning those who trust their own works of obedience to avoid God’s curse are under actually under the curse of the law. That’s because the only way to be free from the curse of the law is to obey the whole law of God perfectly. No one can do that; therefore, no one should try to obey God’s law to free himself from the curse.
Fourth, to be “under the law” is to seek eternal life by obedience to the law. Paul says in Galatians 3:21, “If a law had been given that could give life, then righteousness would indeed be by the law.” Anyone who relies on his own obedience to the law in order to obtain eternal life is “under the law.” Jesus alone is “the way, the truth, and the life.” Anyone who trusts his own good works as his righteousness for eternal life has rejected Christ and is under the law.
Now it’s important to note that in Galatians, Paul does not teach that faithful Christian obedience to God’s law is the equivalent of being “under the law.” In fact, in a number of places, Paul teaches the necessity of obedience to the law of God. In Galatians 6:2, Paul says, “Bear one another’s burdens, and so fulfill the law of Christ.” In Paul’s letter to the Ephesians, he gives us an idea of what he means by “the law of Christ.” He teaches that those in Christ, under grace, are to keep the Ten Commandments as an expression of their faith. True believers are to “put away falsehood,” the 9th commandment (Eph 4:25), “be angry and do not sin,” the 6th commandment (Eph 4:26), “no longer steal,” the 8th commandment (Eph 4:28), abstain from “sexual immorality,” the 7th commandment (Eph 5:3), refrain from “covetousness,” the 10th commandment (Eph 5:3). A true believer is not to be an “idolater,” which refers to the 1st through 3rd commandments (Eph 5:5). A believer is to “honor your father and mother,” the 5th commandment (Eph 6:2).
Paul may even reference the Lord’s Day, or Christian Sabbath, in Ephesians 5:16-19, when he says, “Making the best use of the time because the days are evil . . . addressing one another in psalms, hymns and spiritual songs.” He speaks of “days” in a section of his letter where he is already expounding the Ten Commandments for the believer. Apparently, there is a day on which Christians are to gather to sing and worship the Lord together. Some say that the New Testament repeats 9 of the 10 Commandments, but that it is silent on the Sabbath commandment. Of course, this is false. The New Testament does repeat the Sabbath commandment. Jesus positively declares, “The sabbath was made for man” (Mk 2:27), and Hebrews says, “There remains a Sabbath rest for the people of God” (Heb 4:9).
Many times, in the New Testament, the moral law of God, the Ten Commandments, are viewed as normative for the believer (Rom 2:21-22, 26, 29; 13:9; 7:25; 8:4; Jas 1:25; 2:8-13). This clearly shows that to be “under the law” is not equivalent to keeping the Ten Commandments under grace in faith, which is “the law of Christ.”
Under the Law
1. Consider the meaning of the phrase “under the law” in Pauline literature.
Perhaps the best way to understand the phrase is to study the Galatian heresy. Paul used the phrase “under the law” five times in his letter to the Galatians (3:23; 4:4; 4:5; 4:21; 5:18), more than any of his other letters. But how were the false teachers in Galatia attempting to bring God’s people back “under the law?”
First, to be “under the law” is to attempt to secure God’s verdict of justification by the law. Paul says in Galatians 2:21, “I do not nullify the grace of God, for if justification were through the law, then Christ died for no purpose.” The false teachers were teaching Christians that they had to keep the whole law of God for their justification, not only the Ten Commandments, but circumcision and the Jewish food laws and festivals. This was a false gospel (Gal 1:6-7).
Second, to be “under the law” is to attempt to obtain the Spirit through obedience to the law. Paul says in Galatians 3:2, “Let me ask you only this: Did you receive the Spirit by works of the law or by hearing with faith?” Thus, to seek not only the objective blessing of justification by the law, but also the subjective blessing of the Spirit by the law is to be “under the law.” The Judaizers were teaching that to obtain the Spirit of Christ, one must obey the law of God. That is a “bewitching” heresy.
Third, to be “under the law” is to “rely on the works of the law.” Paul says in Galatians 3:10, “For all who rely on works of the law are under a curse; for it is written, ‘Cursed be everyone who does not abide by all things written in the Book of the Law and do them.’” Thus, all who rely on the law, meaning those who trust their own works of obedience to avoid God’s curse are under actually under the curse of the law. That’s because the only way to be free from the curse of the law is to obey the whole law of God perfectly. No one can do that; therefore, no one should try to obey God’s law to free himself from the curse.
Fourth, to be “under the law” is to seek eternal life by obedience to the law. Paul says in Galatians 3:21, “If a law had been given that could give life, then righteousness would indeed be by the law.” Anyone who relies on his own obedience to the law in order to obtain eternal life is “under the law.” Jesus alone is “the way, the truth, and the life.” Anyone who trusts his own good works as his righteousness for eternal life has rejected Christ and is under the law.
Now it’s important to note that in Galatians, Paul does not teach that faithful Christian obedience to God’s law is the equivalent of being “under the law.” In fact, in a number of places, Paul teaches the necessity of obedience to the law of God. In Galatians 6:2, Paul says, “Bear one another’s burdens, and so fulfill the law of Christ.” In Paul’s letter to the Ephesians, he gives us an idea of what he means by “the law of Christ.” He teaches that those in Christ, under grace, are to keep the Ten Commandments as an expression of their faith. True believers are to “put away falsehood,” the 9th commandment (Eph 4:25), “be angry and do not sin,” the 6th commandment (Eph 4:26), “no longer steal,” the 8th commandment (Eph 4:28), abstain from “sexual immorality,” the 7th commandment (Eph 5:3), refrain from “covetousness,” the 10th commandment (Eph 5:3). A true believer is not to be an “idolater,” which refers to the 1st through 3rd commandments (Eph 5:5). A believer is to “honor your father and mother,” the 5th commandment (Eph 6:2).
Paul may even reference the Lord’s Day, or Christian Sabbath, in Ephesians 5:16-19, when he says, “Making the best use of the time because the days are evil . . . addressing one another in psalms, hymns and spiritual songs.” He speaks of “days” in a section of his letter where he is already expounding the Ten Commandments for the believer. Apparently, there is a day on which Christians are to gather to sing and worship the Lord together. Some say that the New Testament repeats 9 of the 10 Commandments, but that it is silent on the Sabbath commandment. Of course, this is false. The New Testament does repeat the Sabbath commandment. Jesus positively declares, “The sabbath was made for man” (Mk 2:27), and Hebrews says, “There remains a Sabbath rest for the people of God” (Heb 4:9).
Many times, in the New Testament, the moral law of God, the Ten Commandments, are viewed as normative for the believer (Rom 2:21-22, 26, 29; 13:9; 7:25; 8:4; Jas 1:25; 2:8-13). This clearly shows that to be “under the law” is not equivalent to keeping the Ten Commandments under grace in faith, which is “the law of Christ.”