Does a commander's color identity include the colors of activated ability costs in the text box?

A card's color identity is determined by looking at every mana symbol that appears anywhere on the card: its mana cost, its color indicator, and every mana symbol in its rules text, including those in activated ability costs, triggered ability costs, and reminder text (CR 903.4).

This means if a card is colorless or mono-colored in its casting cost but has an activated ability that costs {R} in its text box, red is part of its color identity. You cannot include that card in a Commander deck unless your commander's color identity also includes red.

The same rule applies to symbols appearing in characteristic-defining abilities or any other part of the rules text — all mana symbols count toward identity. Note that generic mana symbols like {1} or {X} do not contribute any color to the identity (CR 903.4a).

Concrete example: Crimson Kobolds is a colorless creature with no mana cost, but imagine a card like Spectral Searchlight that has an activated ability costing {1} to produce colored mana — because the mana symbol in the cost is generic, it adds no color. Contrast this with a card like Opal Palace's older analogues: if a card's activated ability cost included {G}, green would be part of its color identity even if its casting cost were entirely white.

Bottom line: always scan the entire text box for colored mana symbols when evaluating a commander's color identity. Any colored symbol in an activation cost locks that color into the identity (CR 903.4, 903.4a).