@Matt Zaremsky I think that would do it then. If $x_0 \in \langle \langle w\rangle \rangle$ and $w$ is cyclically reduced, we know from the recent discussion (mathoverflow.net/questions/424715/…) that $x_0$ cannot be a proper subword of $w$. So $w=x_0^{\pm 1}$. A very useful observation indeed!

It is true that this is a basis., therefore each element is primitive. But I don't think it provides an example. I assume you are taking $x_0 = z$. In this case, after applying $\pi$ to your set, we get $\{ xyx^{-1}y^{-1},xy,y \}$.