Can your commander's color identity include colors from activated abilities with colored mana costs?

A commander's color identity is determined by every colored mana symbol that appears anywhere in its rules text, including in the costs of activated abilities. This is established in CR 903.4, which states that a card's color identity includes the colors of any mana symbols in its text box, not just its casting cost.

For example, if a commander's activated ability costs {R} to activate (such as a tap ability that reads '{R}: Do something'), then red is part of that commander's color identity even if the card itself has a white casting cost. Your deck would be permitted to include red cards as a result.

This rule also extends to loyalty abilities on Planeswalker commanders, triggered ability costs, and even reminder text — though reminder text in italics is explicitly excluded by CR 903.4d. Mana symbols inside flavor text are also excluded.

Activated abilities with hybrid mana symbols or Phyrexian mana symbols in their costs also contribute both (or all) possible colors from those symbols to the commander's color identity, per CR 202.2c which defines how hybrid and Phyrexian symbols are treated for color identity purposes.

So when building your deck, always scan the full text box of your commander — every colored mana symbol in every ability, including activated ones, expands the color palette of cards you are allowed to include.