Why can't Alphonse trade his alchemy for Edward's alchemy? At the end of the FMA: Brotherhood (Fullmetal Alchemist) series, Edward tried to use alchemy and failed because he wanted his brother back and the law of equivalent exchange clearly states that


  Humankind cannot gain anything without first giving something in return. To obtain, something of equal value must be lost.


Therefore Alphonse's alchemy would be an equal value to Edward's alchemy. so my question is if Edward Elric traded his brother's (Alphonse Elric) body and soul with his alchemy, why can't Alphonse trade his alchemy for Edward's alchemy?

Previously, it was asked Why can&#39;t Edward get back his power to do alchemy? That question was (in my perspective) saying Ed should use a philosopher's stone to do alchemy, because a philosopher's stone ignores the law of equivalent exchange. My question focuses on the observation that a transmutation by Al to give up his alchemy for Ed's should be a good example of following the law of equivalent exchange.
This is an expansion of my thoughts in Why can&#39;t Edward get back his power to do alchemy?

There is no explicit indication that this is impossible, but it would not be in keeping with the tone of the manga (which Brotherhood roughly adapts) were Edward able to recover his alchemy. Edward's decision to trade his gate for Alphonse is portrayed as rather final, per Truth's question to him after he proposes the exchange:


  The Gate of Truth exists within everyone. That is how everyone can use alchemy. Will you degrade yourself to an ordinary person, unable to use alchemy?


Brotherhood points out more explicitly:


  Are you sure about this? If you lose your gateway, you'll never be able to use alchemy again.


It is thus unlikely that the exchange can be reversed; Edward even notes that his recovery of Alphonse is "the Fullmetal Alchemist's final transmutation." Furthermore, the personal nature of the gate makes me doubt that it is possible for someone else to make an exchange for Edward's own gate. (If Edward is the only person who can access his gate and thus, be able to give it away, then he is probably the only person who would be able to exchange it back, in the scenario where regaining one's alchemy is possible.)

Aside from the issue of whether the exchange is technically possible, Edward also does not really care about regaining his lost abilities. Even if he mourned the loss, it is doubtful that he would let Alphonse sacrifice himself in this manner. (Alphonse notes after the time-skip that Edward does not mind keeping his automail leg "as a reminder" and that "apparently he had no intention of trying to get it back" &mdash; Edward only really cared about recovering Alphonse's body. Surely Edward does not care about being "an ordinary person" who cannot use alchemy, if he is even willing to live with his automail leg.)