Core aspects: Consensual, non-destructive, event-only, friendly, with the actual sparring outside of Mobius of course.
After a nice match with another Mobius Commander, of course in a different Private Group, I thought it could be nice to have a "PvP sparring club": Just gathering for set or spontaneous PvP sparring sessions where not the destruction of the other Commander, but fun and skill honing are the focus. Thus disengaging at 50% hull or whatever the duellants agree upon. Note that your ship might still be destroyed due to accidential ramming and such, but everyone does his/her best to avoid destruction of fellow Commanders.
Why this?
PvP-dedicated groups are often very focused on the training aspect, and try to get the most out of their ships for engagements. Trainings are (more or less) tightly evaluated. You learn a lot, but it is, well, like a lesson (which is all fine, yet not our goal here).
Through Mobius, we have the chance to meet other Commanders with the same "sparring for fun"-attitude in mind.
Others like Canonn do PvP events now and then, and everyone just flies whatever he/she has, which I experienced as quite a happy affair. So why not offer the opportunity for some duels, also for Mobius members? Even though we focus on PvE, a PvP session can be fun -- it is much different to fly against another Commander or even a Commander Wing than against NPCs.
At the same time, we still keep the Mobius spirit up by explicitely doing only staged duels, non-griefer and non-accidential, and by switching to another Private Group during the event.
How does this sound?