Boolean Group is Abelian/Proof 2

Theorem
Let $G$ be a Boolean group.

Then $G$ is abelian.

Proof
Let $ a, b \in G$.

By definition of Boolean group:
 * $\forall x \in G: x^2 = e$

where $e$ is the identity of $G$.

Then:

Thus $a b = b a$ and therefore $G$ is abelian.