The only thing Axel's eliminated is vexen.
Under Marluxia's order, Axel came to Twilight Town's replica world and stabbed and defeated Vexen after Vexen disobeyed proper orders. Axel wasn't on Marluxia's side, but he needed to gain Marluxia's trust to succesfully prevent Marluxia's eventual betrayal of the Organization.
Axel destroyed himself to save Sora while in Bixtwixt & Between. Axel recognized Sora as the somebody of Roxas, so he felt the need to help him. When Sora was trying to get to The World That Never Was by using a passage way in Twilight Town, he ended up in a path known as Betwixt & Between. Sora found himself surrounded by Nobodies, so Axel turned himself into a flurry of dancing flames, ending his existence and saving Sora.
Axel Lieber has written: 'Axel Lieber'
Axel Coon's birth name is Axel Broszeit.
Axel Hedfors's birth name is Axel Hedfors.
Axel Springer's birth name is Axel Csar Springer.
Axel Zuber's birth name is Axel Andr Zuber.
Axel Uriegas's birth name is Axel Uriegas Duarte.
Axel Vera's birth name is Axel Lott Vera.
Axel Bergman's birth name is Axel Leonard Bergman.
Gabriel Axel's birth name is Gabriel Axel Mrch.
Axel Axgil's birth name is Axel Lundahl-Madsen.