A monoid where square of all elements are 1 is abelian The following problem gives me a very hard time:

Let $M$ be a monoid with $a^2 = 1$ for $a \in M$. Show that $M$ is abelian.

It looks so simple as a monoid only needs to be associative and must have a neutral element (here $1$). So there are not much things to try. However, after some hours of trying I have to admit that I don't know what to try next.
Maybe somebody of you can give me a hint.
Kind regards!

$1=abab\to b=ababb \to b=aba \to ab =aaba \to ab =ba $
