Kevan, the Earl of Io, may challenge Elodie to a duel to the death. If Elodie has sufficient mastery of her Lumen skills, she can instantly burn him to a crisp while delivering a Bond One-Liner.
On that note, if your skill in Swords is high enough, you can accept his challenge, and then beat him in a fair fight.
Alternatively you can fight him with a staff and beat him. Kevan will be so out of himself he'll try to break the rules and force you to strike him down. You can talk him out of it with enough of the right set of skills, as the game itself puts it: "The applause of the crowd is deafening."
Yet another option consists of training your falconry skill to max level, then choosing the Falconer contest. When Kevan shows up, your falcon will swoop in and hack away at him like the true BadassAction Pet it is until he's detained. Remember that Kevan wears plate armour and is almost impossible to kill with magic.
To be precise, you defeat Togami by using the sole area of combat Togami isn't an expert in: a physical attack. You create a sword made out of magic and cut off Togami's head.
And an understated (and very tricky-to-get) version is if you manage to get him to stop the war by singing to him, then appealing to him to think of his children, and what might happen if they lost their father.
The epilogue for this one reinforces it - among the gifts the Shanjian trade-ship brings to Elodie, there's two paintings of "a pink-haired angel" obviously made by children.
Prior to the duel, it's possible to actually drive off the Shanjian fleet altogether, preventing their army from even making it to your shores, making Elodie very popular amongst the commoners. One way of doing this is to utterly obliterate it with magic. Pity it summons the Kraken, but it's still terrifyingly awesome.
Even without obliterating them in a magical ritual, you can end up with a tiny pink-haired girl perched at the top of a crow's nest, bellowing insightful and intelligent orders to her fleet while she launches fireballs of magic at any enemy ships that dare to come in range, until the enemy fleet limps back home in disarray. It's pretty great.
The 2015 plot update has an option to consult your magical tutor for help if you sense that the man who killed his wife has a magical aura, but you can't sense further details. If your tutor is Julianna, she recognizes him as a Lumen, and ends up defeating him in a magical battle in front of the entire court. With the right skills, Elodie can immediately turn to the watching nobles and deliver a spur-of-the-moment speech on how this exemplifies the need for loyal Lumens in the royal service. The fight is a moment for Julianna, the speech for Elodie, and between the two, the game explicitly says that you'd probably get a round of applause if it wouldn't violate etiquette.
An understated moment when Elodie discovers another Lumen (and unknowingly heads off some nasty assassination attempts).
Lucille: Look at how much trouble the Duchess of Ursul has. Please, Your Highness, don't tell anyone.
Elodie: What I think I should tell everyone is that you are my new Lumen Minister.