Let $R$ be a subset of $\text{PSL}_2(\mathbb C)$ and consider its natural action on $\mathbb {CP}^1$. We say that $R$ is elementary if either $R$ is conjugated to a subset of $\text{SU(2)}$ or if there is a finite subset of $\mathbb {CP}^1$ which is invariant by every element of $R$.

For $n \geq 1$ set $R^n : = \{g_1\cdots g_n : g_i \in R\}$ (that is, the set of products of elements in $R$ of length $n$) and $R^n \cdot R^{-n}:=\{gh^{-1} : g,h \in R^n\}$. I'm looking for a reference for the following statement, which seems to be true:

*Assume $R$ is non-elementary. Then there exists an $N \geq 1$ and an $M \geq1$, such that $R^{N}$ contains a loxodromic element and $R^{M} \cdot R^{-M}$ contains a non-elliptic element.*

I have a somewhat tedious proof of this fact but I would guess that such a result is well-known (at least the one about $R^N$). A reference to a book or a paper containing these results or closely related ones would be of great help.

Thanks in advance for your answers.