Does Commander damage count if the commander was given double strike?

Commander damage tracks all combat damage dealt by a commander to a player over the course of the game. When a creature with double strike deals combat damage, it deals damage in both the first strike damage step and the regular combat damage step. Both instances are real combat damage dealt by that commander, so both count toward the 21-point threshold (CR 903.10).

Double strike is defined in CR 702.4b: the creature assigns damage in the first strike combat damage step AND the regular combat damage step. Neither hit is excluded from commander damage tracking — the rule simply says "combat damage dealt" with no exceptions for how it was delivered.

This means a commander with power 11 and double strike can deal 22 commander damage in a single attack (11 in first strike, 11 in regular), instantly eliminating a player who has taken none previously, since 22 exceeds the 21-damage threshold.

Concrete example: Your Isshin, Two Heavens as One (a 3/4 commander) is granted double strike via Tempered Steel effects or a spell like Duelist's Heritage. It attacks and connects — it deals 3 damage in the first strike step and 3 damage in the regular damage step. That player has now taken 6 commander damage from Isshin this game. Accumulate 21 total and they lose.

Note that commander damage is tracked per commander and is cumulative across the entire game, regardless of how many times the commander has changed zones (CR 903.10a). Double strike simply accelerates how fast that total is reached.