My opponent steals my Skyfall with Akshan, but I play another Akshan to take it back - what happens when my Akshan dies?

When your Akshan leaves the board, control of the Skyfall of Areion will revert to your opponent, not you.

Here is the step-by-step breakdown of the interaction based on the rules of control-change effects and the precedents established in RiftJudge FAQs #8851 and #5396:

1. Initial Theft: Your opponent plays their Akshan and uses its ability to steal your Skyfall of Areion. Their Akshan creates a control-change effect that grants them control of the gear for as long as their Akshan remains on the board.
2. Your Counter-Theft: You play your own Akshan and use its ability to take control of the Skyfall of Areion. Your Akshan creates a new control-change effect. Because your effect is the most recent, it overrides your opponent's effect due to timestamp order. You now control the gear.
3. Your Akshan Dies: When your Akshan leaves the board, the control-change effect it created ceases to apply.
4. Recalculation: With your control-change effect gone, the game recalculates control. Since your opponent's Akshan is still on the board, its original control-change effect—which was previously overridden—becomes active again.
5. Result: Control of the Skyfall of Areion reverts to your opponent, as they are the controller of the only remaining active control-change effect.

Important Nuances:

• Attachment: As noted in RiftJudge FAQ #8714, the gear does not automatically detach or return to your base when control changes. It remains attached to whatever unit it was on (or stays at your base if it was unattached).
• Ownership vs. Control: While your opponent regains control, you remain the owner of the Skyfall of Areion (the player who brought it into the game). If the gear were to be returned to a hand (e.g., via Factory Recall), it would return to your hand, not your opponent's, per Rule 107.6.f and RiftJudge FAQ #9096.