Here is a proof which relies on a straighforward generalization of the [Ping Pong Lemma](https://en.wikipedia.org/wiki/Ping-pong_lemma).

> **Claim.** Let $a$ and $b$ be the transformations of the Riemann sphere $\hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$ defined by 
$$
a(z) = \frac{1}{z} + 2,\quad b(z) = z + 4.   
$$
Let $d \in \mathbb{N}_{> 0} \cup \{\infty\}$. Then the subgroup of $\operatorname{PSL}(2, \mathbb{R})$ generated by $X_d = \{b^{i}ab^{-i} \,\vert\, 0 \le i  < d\}$ is free on $X_d$.

The claim and its proof are a trivial adaptation of [1, Example II.B.26] which makes use of [Schottky groups](https://en.wikipedia.org/wiki/Schottky_group).

> *Proof of the Claim.* Let $\mathbb{D} = \{ z \in \mathbb{C} \, \vert \, \vert z \vert \le 1\}$, $\Omega_i = (\mathbb{D} + 4i) \cup (\mathbb{D} + 4i + 2)$ for $0 \le i < d$. Observe that $\gamma_i = b^ia b^{-i}$ maps 
> - the exterior of the disk $\mathbb{D} + 4i$ onto the interior of the disk $\mathbb{D} + 4i + 2$, and  
> - the exterior of the disk $\mathbb{D} + 4i + 2$ onto the interior of the disk $\mathbb{D} + 4i$.
> 
> Thus $\gamma_i^k(\Omega_j) \subseteq \Omega_i$ for every $i \neq j$ and every $k \in \mathbb{Z} \setminus \{0\}$. Clearly $\Omega_i \nsubseteq \Omega_j$ if $i \neq j$. A straightforward generalization of the [Table-Tennis Lemma](https://en.wikipedia.org/wiki/Ping-pong_lemma) yields the result.

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[1] P. de la Harpe, "Topics in Geometric Group Theory", 2000.