I don't have experience with running a game server specifically, but since being a StackExchange moderator I've learned some tips for dealing with problematic users (both from direct experience on this site, and from hearing how mods on other sites handled issues).
Step 0 is always: when in doubt, talk to the other mods. You all need to be on the same page and be able to present a unified front. Inconsistent enforcement of rules is frustrating and confusing for your community (plenty of comments and meta posts here about that!), and at worst acts as a sort of intermittent reinforcement for bad actors to keep trying.
Next, some advice I got about writing mod messages was never make threats. As Yoda says, do or do not: There's no point in saying "we'll let it slide now, but maybe later we won't!" - if anything, it just encourages them to push the limits, and makes it harder for you to actually act because, well, last time you didn't do anything, what makes this time different? In addition, it can be hard to mark well-defined lines of what is "bad enough". Whether or not a user cheats is probably clear-cut, but rude behavior often involves context and some amount of moderator discretion.
So, don't speculate about possible future actions in this message: simply state what, if anything, you are doing now, and what you need them to do.
In addition, there's no need to explain how you figured out his alter-ego. He already knows you're right, and you don't want to inadvertently give him any ideas about how to better cover his tracks in case he reverts back to his old ways. (This is also something that's notably absent from the StackExchange template for socks; many users have tried to poke and get more details in response to a suspension, but it's intentionally kept secret.)
So, I would message him privately and simply say something along the lines of "we're aware of your past, here's a reminder of the rules, please follow them". For example:
We've noticed you're using a new account after your previous account earned a ban. We appreciate your effort to be a better member of the community, and in light of this improvement, have decided to allow this account to remain on the server.
However, there were some serious disruptions to the community in the past. So, this is just a friendly reminder of the server rules: (link or summarize your policies here)
We look forward to your continued positive participation!
Thanks, The Mods
Of course, adapt to your own style! This is fairly formal-sounding; best to write however you would write other "official" type posts for your server.
If possible, share a draft of your message to the other mods before messaging the user, to again make sure you're all on the same page. The IPS mod team often does this with messages (among other things), which has a few benefits:
- Everyone has a chance to give input and tweak language until we're all satisfied
- If the user objects, any one of us is able to handle it (especially useful if there's likely to be real-time response needed)
- Easier to be consistent if a similar situation arises later, since we've already thought this through together
It's also a good idea to discuss now what you'll do if he starts acting badly again. You don't need to tell him what those tentative lines are, but having a plan in place makes it easier to act if it happens again.
Good luck - it's generous of you to give him a second chance, and I hope it's a lasting change for the better!