In what way was Jesus' command in John 13:34 "new"?
34 A new commandment I give to you, that you love one another. Just as I have loved you, you also love one another. John 13:34WEB
It isn't the "love one another" part that is new -- that was already commanded back in Lev. 19:18. The new part is "as I have loved you." We have now seen YHWH in the flesh interacting directly with us and providing us with specific examples of how we should love. I think a lot of these examples involve Jesus' specific emphases on humility, care for children, and personal forgiveness. But he also exemplified following Biblical law, as well.
So the next question should be "did Jesus provide us with any examples of 'loving action' which overturn or contradict anything that was previously commanded in God's law?"
I suggest the answer must be no. Jesus violated (often intentionally) the Pharisees' traditions about the law, but Jesus never violated the law itself. Rather, he upheld the law in ways that made men uncomfortable. Therefore, he could never have given us an example of an action which overturns the law, only actions which clarify it, or contradict men's additions to (or traditions about) the law.
- 18... but you shall love your neighbor as yourself. I am YHWH. Leviticus 19:18WEB