[RPG] Is it possible to join multiple factions

Use your character's backgrounds to build intrinsic motivation for your players' characters.

If you're looking at the LMoP adventure, you hopefully have the pre-built characters handy. If you look at their backgrounds, you can see that for every character, there is some kind of intrinsic motivation built into their backgrounds for at least one of the elements of the adventure.

This provides the characters with clear motivation for why they should care about both the mission and the results of the mission.

You can do this in a custom setting in a couple of different ways. First, you could assign different background elements to your players, this may be a bit of a deal breaker for some players, but I've had good luck with it using a pre-built adventure and giving my players the option of which of 10 or so to select (Horde of the Dragon Queen provides 10 different optional background elements).

However, the best thing you can do is to pick up plot elements from your characters' backgrounds. Have them write a short paragraph or even just be super detailed when they are building their BIFT part of their background so that you can mine that for details and NPCs and other motivations.

If you're looking to provide intrinsic motivation, having it worked into your characters' backgrounds is the best and easiest way to do it.

I think the one of the best solutions would be to ask your DM about that. You can work with him to create a group of wizards that fits your needs.

Though you are playing in the particular setting of the Forgotten Realms, there is nothing that forbids the creation of another group. The minor drawback is that it is extra work for the DM.