Let $I=\left]-1,\ 1\right[$ be an interval, and $\left(I,\ \star\right)$ be a magma such that:
$$\left(\forall\ \left(x,\ y \right) \in I^2\right)\ x \star y=\frac{x+y}{1+xy}$$
I need to prove that $\left(I,\ \star\right)$ is an Abelian group.
A simple way to prove it is by checking that the magma $\left(I,\ \star\right)$ satisfies the group axioms including commutativity.
Based on the following statement I would like to approach that problem:
Let $f$ be a homomorphism from $\left(X,\ \perp \right)$ to $\left(I,\ \star\right)$, then the algebraic structure of $\left(I,\ \star\right)$ is exactly the algebraic structure of $\left(X,\ \perp \right)$
So I would like to find out a usual Abelian group - ex. $\left(\mathbb{R},\ +\right)$ - and a homomorphism $f$ from that group to $\left(I,\ \star\right)$.
Assume that $f$ is a homomorphism from $\left(X,\ \perp \right)$ to $\left(I,\ \star\right)$, then
$$\left(\forall\ \left(x,\ y \right) \in X^2\right)\ f\left(x \perp y \right)= f\left(x\right) \star f\left(y \right)$$
$$\Leftrightarrow f\left(x \perp y \right)= \frac{f\left(x\right) + f\left(y \right)}{1+f\left(x\right) \cdot f\left(y \right)}$$
I got stuck on that. Could anyone support me with some hints how to get it done.
Hint: $$\tanh (a+b)=\frac{\tanh a+\tanh b}{1+\tanh a\tanh b}.$$