By recent chapter, I think Dowun have no right to be mad/jealous that Hong Lyon has shifted her affection to Baek Han. When they're younger, Dowun is all Hong Lyon ever have and she outwardly shows that. In return, what does Dowun give her emotionally? Contempt, and all things that you don't want to feel. Dowun made it very clear that she's just one of the many human tools he have. And while Dowun doesn't mistreat Hong Lyon, he's not exactly the nicest to her. Dowun is very good at differentiating between personal and business relationship, that's what he teaches Hong Lyon too, so idk why he suddenly want to mix them together now?
In earlier chapter, Hong Lyon basically asked that going by Dowun's principle, would it mean that he doesn't mind if she loves another even while married to Dowun? And his answer is yes. How would Hong Lyon keeps her affection/love/ to him when he's showing disinterest in her own feeling like that? I'd think that the last bit of her romantic feeling (and emotional attachment) for Dowun got thrown away along with their engagement ring, and now her pairing earrings.
While I think that Dowun does love Hong Lyon to an extent, for some reason he isn't showing the love she deserves. He's maybe held back by his personality, upbringing, responsibility or the weight of Dowun's family name or even those stuffs altogether. But you see, it takes two for a relationship. It can't work if Dowun can't show even in the least affection for Hong Lyon. Because from what the author has given us to see, Hong Lyon never feel Dowun's affection for her. Hell, we only have Dowun's jealousy outburst as a basis that Dowun even love her (and maybe some other really small hints dropped along the way). For now, I think that Dowun's sudden burst of jealousy comes from the fact that Hong Lyon is no longer an effective tool because of her affection for Baek Han. But really, it could also be that this is the result of Dowun trying really hard to not fall for her (esp when they're younger), because it'll make her weak and make him weak too, and weakness is something that others could exploit, in imperial castle especially. I still don't know at this point of time, my judgement for Dowun is still towards the negative side, so we'll know more when the author progress with the story more.
But it's also right that without Dowun, Hong Lyon will not gain the power she has now. However small that power is, because as we all see, the real power is behind her aka Dowun and his dad. Dowun is the one who teaches her the basic of all basic, and maybe it wouldn't be exaggerating that Dowun's the one who made her worthy. But looking from another side, it's also seen that Hong Lyon is actually talented and a genius that raised the question, wouldn't Hong Lyon also blooms and raised merit even if the one who teaches her is not Dowun? While it's only mentioned in passing, Hong Lyon was sent to another country alone, without Dowun, and she's able to get by there. Their contact would only be by messages, and I assume it wouldn't be much going by Dowun's personality and attitude when he welcome her back.
For now, I think the pairing would be Baek Han/Hong Lyon, Dowun's lil bro/Hong's maid and one sided Asara/Dowun.
Relationship wise, Dowun/Hong Lyon isn't healthy but as much as I like Baek Han/Hong Lyon, I would really support Dowun/Hong Lyon if they could talk together and get things straight. Right now, Hong Lyon is really easily swayed around by Baek Han too so I hope she could show more growth to her character. I hope Dowun could express his emotion more too, at least in front of Hong Lyon. He really needs to talk more to Hong Lyon, I think that she would actually like it more if Dowun could give her more job so she can feel more responsibility. Not giving her the stuffs she should be working is like saying they didn't believe in her ability to give judgement, hence why she isn't touching the work that Dowun has done in her place. Positive thinking, Dowun may think he's helping her (he is! Helping her) but idk what Hong Lyon feel when she's really just a figurehead that can be replaced by Dowun as the emperor at this point.