Lieutenant abilities apply only if your commander is on the battlefield and under your control.

If you lose control of your commander, lieutenant abilities of creatures you control will immediately stop applying. If this causes a creature’s toughness to become less than or equal to the amount of damage marked on it, the creature will be destroyed.

Lieutenant abilities refer only to whether you control your commander, not any other player’s commander.

If a triggered ability granted by a lieutenant ability triggers, and in response to that trigger you lose control of your commander (causing the lieutenant to lose that ability), that triggered ability will still resolve.

If you gain control of a creature with a lieutenant ability owned by another player, that ability will check to see if you control your commander and will apply if you do. It won’t check whether its owner controls their commander.