In Commander, does a face-down commander in the command zone still have its color identity?

Color identity is calculated from a card's printed characteristics — its mana cost, color indicator, and any mana symbols in its rules text — before the game begins. It is not a characteristic that changes based on game states like being face-down. This is established by CR 903.4, which defines color identity, and CR 903.3, which governs deck construction using that identity.

Importantly, the command zone is not the battlefield. Face-down rules (CR 707) only apply to permanents on the battlefield. A card sitting in the command zone is never a permanent and can never be face-down in any meaningful game-state sense — it simply exists there as a card with its full printed identity.

Even if a future mechanic allowed a commander to be 'face-down' in the command zone, CR 903.4 specifies that color identity is determined from the card's characteristics as printed, so it would still contribute its full color identity to deck-building restrictions and color-related effects like Chromatic Lantern or color-checking spells.

Concrete example: If your commander is Yidris, Maelstrom Wielder (UBRG color identity), it retains that four-color identity in the command zone no matter what. Your 99-card deck must still conform to those four colors. There is no scenario in Commander where being 'face-down' in the command zone strips away that identity.