When Sylph asked Lake if she could sit next to him, Lake felt that she was purposefully causing trouble.
It would have been fine if she had just walked up and sat next to him, at least that way it would seem she just wanted to sit in the back but instead she asked if she could sit next to him.
This may seem like she was being polite, but to everyone else the way she worded it was asking for trouble.
Instead of asking if she could sit in the seat next to him, she directly asked if she could sit next to him, implying that she wanted to be close to Lake and wasn't after the seat.
Even if that wasn't her intention and she was just bad with words, it's too late now, the damage has been done.
Feeling the hate filled gaze of everyone in the class, Lake felt a headache.