Hey, i just came with a theory what if the principal/teacher (i don't know what she is) is a bear, but not just a bear, is the bear that ate kureha's mom
even though she has yuri in her name, i dont think the teacher (yuriika) is a bear. reason for that being that all the bears have yuri written as 百合 while the teacher's name is written with ユリー. there might be some significance in the reason why the bear names use 百合 but the teacher, the yuri court, and the title of the show use ユリ.
however, there seems to be similar themes/situations between YKA and Penguindrum. and because these similarities exist, i also think that yuriika killed kureha's mom (reia). its kinda similar to the situation in penguindrum where spoilers for penguindrum Yuri's jilted lover from the acting troupe stabbed Tabuki in the abandoned restaurant. end spoilers so i think that reia and yuriika had a thing going on, but yuriika had her heart broken in a cruel fashion. after all, reia had to get pregnant somehow. yuriika lost it, and if she cant have reia, no one can.
YKA is based on the Sankebetsu brown bear incident. one of the victims was a pregnant women who ended up begging the bear to eat her head in an attempt to save her child. i think kureha's family situation is meant as a reference to this incident. because whenever kureha gets knocked into yuri court by the bears, she falls while in the fetal position.
because of this, i believe the weapon yuriika used to kill reia is bear gingko. during the attack, reia begged gingko to spare her child (kureha), and she does. gingko keeps reia's choker because it reminds her of how delicious reia was and hopes that kureha tastes the same. yuriika ends up caring about kureha because even though reia broke her heart, yuriika still loves reia and kureha is the last connection she has to reia.
that connection would make another parallel spoilers for penguindrum between yuriika and yuri. yuriika cares for kureha because she reminds her of reia. similarly, yuri cares for ringo because she reminds her of momoka. end spoilers
hell, im starting to think yuriika is the mastermind behind everything. the name show and the yuri court have her name in it. she uses her court to see if the girls have the same feelings she had for reia and uses the invisible storm to deny them what the world denied her. and there's the birds on the railings that always go in the direction against the way people are walking matching the bird pin yuriika has on her collar and the bird/lily tile design.