The key is to understand from Pauline theology that the dichotomy of Law vs. Grace applied only to the Mosaic covenantal law, and not the general or moral law. This understanding helps us solving the traditional false dichotomy of "justification by faith alone" with "faith without works being dead." We can only fulfill the grace of Christ by following Christ's law that is his teachings or commandments. Only the Mosaic law has been done away, fulfilled or abolished. Believers still remain under the law of God, that is the same moral spiritual laws minus the Mosaic covenantal law.
Paul did not felt the need to repeat this same phrase "law of Christ" perhaps he thought he has implicitly taught the law of Christ frequently in his letters. The gospels also teach the law of Christ. See the parables like Matt 25:14-30
Matt 7:12 “So whatever you wish that others would do to you, do also
to them, for this is the Law and the Prophets. (verse) 21 “Not
everyone who says to me, ‘Lord, Lord,’ will enter the kingdom of
heaven, but the one who does the will of my Father who is in heaven.
22 On that day many will say to me, ‘Lord, Lord, did we not prophesy
in your name, and cast out demons in your name, and do many mighty
works in your name?’ 23 And then will I declare to them, ‘I never
knew you; depart from me, you workers of lawlessness.’ ESV
John 15:6 If anyone does not abide in me he is thrown away like a
branch and withers; and the branches are gathered, thrown into the
fire, and burned. (verse) 15:10 If you keep my commandments, you
will abide in my love, just as I have kept my Father's commandments
and abide in his love. ESV
The various explicit teachings of judgment according to our works or
"we reap what we sow" also imply the Law of Christ. See Rev 2:23; 22:11-12; Rom 2:6-7, 14:12; 2Cor 13:5-6, 5:10
Rom 6:19-22 19 I am speaking in human terms, because of your natural
limitations. For just as you once presented your members as slaves to
impurity and to lawlessness leading to more lawlessness, so now
present your members as slaves to righteousness leading to
sanctification. 20 For when you were slaves of sin, you were free in
regard to righteousness. 21 But what fruit were you getting at that
time from the things of which you are now ashamed? For the end of
those things is death. 22 But now that you have been set free from sin
and have become slaves of God, the fruit you get leads to
sanctification and its end, eternal life. ESV
Rom 11:22 Note then the kindness and the severity of God: severity
toward those who have fallen, but God's kindness to you, provided you
continue in his kindness. Otherwise you too will be cut off. ESV
Rom 14:12 So then each of us will give an account of himself to God. ESV
Gal 5:13 For you were called to freedom, brothers. Only do not use
your freedom as an opportunity for the flesh, but through love serve
one another. ESV
Phil 2:12 Therefore, my beloved, as you have always obeyed, so now,
not only as in my presence but much more in my absence, work out your
own salvation with fear and trembling, ESV
Jas 2:8-11 8 If you really fulfill the royal law according to the
Scripture, “You shall love your neighbor as yourself,” you are doing
well. 9 But if you show partiality, you are committing sin and are
convicted by the law as transgressors. 10 For whoever keeps the whole
law but fails in one point has become accountable for all of it. 11
For he who said, “Do not commit adultery,” also said, “Do not murder.”
If you do not commit adultery but do murder, you have become a
transgressor of the law. ESV
1Pet 1:15-17 15 but as he who called you is holy, you also be holy in
all your conduct, 16 since it is written, “You shall be holy, for I am
holy.” 17 And if you call on him as Father who judges impartially
according to each one's deeds, conduct yourselves with fear throughout
the time of your exile, ESV
1Pet 2:16 Live as free people, not using
your freedom as a pretext for evil, but as God’s slaves. NET