A New Commandment which is from Old

"Those that are indeed commandments from Christ are not new commandments that have been given by Christ that a condition of justification might be established in them. But they are old, restored by Christ, not only through the explication of the law not sufficiently explained and observed by the Pharisees, but also through its fulfillment. Certainly this fulfillment makes not only for the acquisition of Christ the true Mediator, to stir up our prayers, that we may more ardently entreat the Spirit who sanctifies from God, and finally that the understanding of the law may be the means of our sanctification (for the Spirit does not inscribe on our hearts that we may not understand that which He inscribes) and so our holiness may be more true and complete. Therefore, those commandments are old, because they have been restored in the example of Christ and engraved on the hearts of believers by the Holy Spirit (John 13:34). In 1 John 2:8 that is emphatic, 'Again a new commandment I write to you, which is true in Him and in you, because the darkness,' not only of servitude to the shadows but also to hardening, which existed as long as God deserted the Gentiles, 'is passing away, and the true light,' in the word and example of Christ and in the regeneration of the Spirit, 'is already shining.' This pertains to the explanation of the first, second, third, and fifth good of the New Testament. From this it is clear that following Christ (which also comprehends denial of oneself and bearing the cross) is not a new commandment peculiar to Christ, as the Racovian Catechism (on the prophetic office of CHrist, ch. 2) wishes, because the obedience of Christ to be imitated by us is the observance and fulfillment of the Old commandment." - Johannes Cocceius, The Doctrine of the Covenant and Testament of God, Pg. 254

A new commandment I give unto you, That ye love one another; as I have loved you, that ye also love one another. - John 13
Beloved, let us love one another: for love is of God; and every one that loveth is born of God, and knoweth God. 8 He that loveth not knoweth not God; for God is love. 9 In this was manifested the love of God toward us, because that God sent his only begotten Son into the world, that we might live through him. 10 Herein is love, not that we loved God, but... that he loved us, and sent his Son to be the propitiation for our sins. 11 Beloved, if God so loved us, we ought also to love one another. 12 No man hath seen God at any time. If we love one another, God dwelleth in us, and his love is perfected in us. 13 Hereby know we that we dwell in him, and he in us, because he hath given us of his Spirit. 14 And we have seen and do testify that the Father sent the Son to be the Saviour of the world. 15 Whosoever shall confess that Jesus is the Son of God, God dwelleth in him, and he in God. 16 And we have known and believed the love that God hath to us. God is love; and he that dwelleth in love dwelleth in God, and God in him. 17 Herein is our love made perfect, that we may have boldness in the day of judgment: because as he is, so are we in this world.18 There is no fear in love; but perfect love casteth out fear: because fear hath torment. He that feareth is not made perfect in love. 19 We love him, because he first loved us. 20 If a man say, I love God, and hateth his brother, he is a liar: for he that loveth not his brother whom he hath seen, how can he love God whom he hath not seen? 21 And this commandment have we from him, That he who loveth God love his brother also. - 1 John 4

We love because he first loved us. A new command is really the old command given to the Israelites but now is fulfilled in Christ Jesus. Therefore, those in Christ are said to love the brethren. Further, transcribing the law as new doesn't have to mean this is a different law (though believers are not under the law in covenant form). The difference between the law in its covenant of works verses it being in the covenant of grace is basically that the law is formally different but not materially different. In other words the law (though the same as of the law in the ot) is no longer condemning for those in Christ, neither do believers keep the law for gaining salvation. We love because he has loved us and has given his life for us. He has not done this for everyone (though the Arminian falsely claims); for Gods love is effective and is powerful to achieve that which he wills.