After reading the answers to this question, a thought popped into my mind: if Karn Liberated's ultimate restarts the game, and declaring one of the Legendary Creatures in your deck as a Commander is a part of the process of starting the restarted game, could you choose a different Legendary Creature from your deck as your Commander for the restarted game?
For instance, if you were originally playing a deck that used Maelstrom Wanderer as your Commander, and you had Intet the Dreamer as one of the cards in your deck, could you declare Intet as your new Commander after Karn restarts the game?
If you can, do you need to respect the color identity of all the cards in your deck when you do so, since it sounds like you don't check deck legality during Karn's restart process? For instance, if the Maelstrom Wanderer deck also included Tanazir Quandrix, could you declare them as your new Commander, even though you have Red cards in your deck?