WHY? "As BLS public keys reside in G2"

[TinyVege]

It is mentioned in the paper that As BLS public keys reside in G, the signature is in G

I want to know why? Or conversely, put the bls public key in G1 and the signature in G2.

I learned that on the bls12381 curve, both are possible. Ethereum 2 chose the latter "the bls public key is placed in G1 and the signature is placed in G2

We can choose to use either group for public keys, as long as we use the other group for signatures: the pairing function doesn't care; everything still works if we swap the groups over. The original paper describing BLS aggregate signatures has public keys in G2G2​ and signatures in G1G1​, while for Ethereum 2 we made the opposite choice.
ref: https://eth2book.info/capella/part2/building_blocks/signatures/#sync-aggregates

There is another idea that I don’t know if it is correct: the pairing operation in e(P1, Q1) = e(P2, Q2) will not change whether the public key is in G1 or G2.

@PhilippSchindler