theHigherGeometer on Nostr: I think it would be nice to identify extra structure such that its stabiliser group ...

I think it would be nice to identify extra structure such that its stabiliser group turns out to be exactly the thing we see (ie the connected group), eg something like the star-algebra structure coming from conjugation, rather than putting it in by hand. Perhaps this is "putting it in by hand" somewhere else, but it's recognising structure that's there, not just throwing out half a group to match experimental data.