Can you still cast a card that was exiled with {Psychic Intrusion} if it's not in your commanders colors? I want to say you can because you're not adding off colors to your mana pool, but would like a confirmation. Thank you :)
I don't think so as colours not aligned with your commander turn colourless and it's a blanket rule you can't cast colours you don't have the mana for.
But I'm not a rules guru. But if it were a part of a {Sen Triplets}/{Mycosynth Lattice}, you could.
This will work. It's because of the whole "you may spend mana as though it were mana of any color" thing. It makes it so you don't have to produce mana outside of your color identity to cast something outside of those colors.
As an example, if your commander is {Grimgrin, Corpse Born} and you exile a green card, you can spend your blue or black (heck, even colorless) mana to cast it, as if it were actually green mana. It's a pretty cool trick.