You know a bit more - first, you make an assumption about the inverse in your question. Then you have just, correctly, written "the" inverse. (As opposed to "some" inverse).
Oh, maybe since because we know that for $a$ the inverse is $-a$ and with how you've shown that $b+a$ is the inverse of $a+b$ in this case we can conclude $a+b -(b+a) =0$?
By assumption, the inverse of any $a\in G$ is $a$. So the inverse of $a+b$ is $a+b$. My calculation shows that also $b+a$ is inverse to $a+b$. What do you now need to finish? Look at my previous comment.
Discussion between Thomas and Lisa
Discussion between Thomas and Lisa