I believe that Tenzin actually made a betrothal necklace, but since we don't see Pema wearing any, I believe he made it for Lin, and since they don't end up together, the necklace now sitting collecting dust inside a box he kept on his study. He can't bring himself to give it to Pema because when he crafted it, it wasn't her in his mind, but he can't make another one either cause it's something he holds sacred. That betrothal necklace was the closest thing he got to marrying the love of his life, yet when he decided to not giving it turned out to be the lost of his life.
So, every once in a blue moon, he would open the box, admiring the necklace and wising it was wrapped around Lin's neck rather than collecting dust in his study.
(ID:
Tenzin: "Today, we welcome our newest airbending master. I know our father would be so proud."
Image of Bumi with arrow tattoos, standing in front of the airbender crest ceremonial hall, mirroring how Jinora also had her ceremony as the music "Service and Sacrifice" plays. He's smiling with tears in his eyes, happy. )
lin has 100% fucked all three kataang siblings. both kya and bumi are aware of this, because they’re not totally oblivious, but poor tenzin still believes he’s the only one.
then, one day, tenzin suddenly has the epiphany during dinner (with his entire family around, including lin and both siblings), and has to try to stay calm while he realises what all the quips made by kya and bumi around him really meant.