Proof that group is commutative if every element is its inverse (feedback wanted)

This is one of my first proofs about groups. Please feed back and criticise in every way (including style & language).
Axiom names (see Wikipedia) are italicised. $e$ denotes the identity element.


Let $(G, \cdot)$ be a group.
We assume that every element is its inverse.
It remains to prove that our group is commutative.
Non-trivially, $\textit{associativity}$ implies that parentheses are unnecessary.
Therefore, we do not use parentheses,
we will not use $\textit{associativity}$ explicitly.

By $\textit{identity element}$, $G \ne \emptyset$.
Now, let $a, b \in G$.
By assumption, $$aa = e \text{ and } bb = e. \quad \text{(I)}$$
By $\textit{closure}$, $ab \in G$.
So, by assumption, $$abab = e.\quad \text{(II)}$$
It remains to prove that $ab = ba$.
\begin{equation*}
\begin{split}
ab &= aeb && \quad\text{by }\textit{identity element} \\
&= aababb && \quad\text{by (II)} \\
&= ebabb && \quad\text{by (I)} \\
&= ebae && \quad\text{by (I)} \\
&= bae && \quad\text{by }\textit{identity element} \\
&= ba && \quad\text{by }\textit{identity element}
\end{split}
\end{equation*}
QED

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