If I steal an enemy equip with a unit ability, but it says the gear is kept until my unit is gone. But I move the gear to a new unit via Jax. Do I lose the gear when the original unit dies?

I cannot provide a definitive ruling on this interaction because the card "Jax" is not found in the Riftbound database, and there is no official FAQ or rule covering the specific interaction of moving stolen equipment between units via a legend ability.

Based on the provided rules and FAQs, I can offer the following analysis:

1. Control of Stolen Equipment

According to RiftJudge FAQ #8562, when a card like Akshan steals your gear, they gain control of that equipment for as long as their Akshan remains on the board. The duration of control is tied to the source of the effect (the Akshan unit), not the unit the gear is currently attached to.

2. Moving Equipment

While the rules for a specific "Jax" ability are not available, it is noted that while a gear's own Equip ability is inactive while attached, external effects (like Weaponmaster or other legend abilities) can be used to move equipment to another unit you control.

3. What happens when the unit dies?

According to Rule 435.1 and RiftJudge FAQ #2960, when a unit dies, any attached gear detaches and remains at the battlefield (or is recalled to the controller's base during the next cleanup).

Interpretation: If your ability to control the stolen gear is tied to a specific unit (like Akshan), moving the gear to a different unit does not change the duration of your control. If the source of the control effect (the unit that stole the gear) leaves the board, you would lose control of the gear regardless of which unit it is currently attached to.

If you can provide the exact text of the card you are referring to as "Jax," I may be able to provide a more accurate assessment based on its specific wording.

Note: I could not fully verify the ruling regarding possessed units against the retrieved rules.