In this paper: https://perso.univ-rennes1.fr/serge.cantat/Articles/nsgc-acta-c.pdf the authors said that their result that $\text{Bir}(\mathbb{P}_\mathbb{C}^2)$ is not simple 'directly implied' that $\text{Bir}(\mathbb{P}_\mathbb{R}^2)$ is not simple as well. How can we infer the non-simplicity of $\text{Bir}(\mathbb{P}_\mathbb{R}^2)$ just from the fact that $\text{Bir}(\mathbb{P}_\mathbb{C}^2)$ is not simple?

I am thinking of choosing a proper normal subgroup, say $N$, of $\text{Bir}(\mathbb{P}_\mathbb{C}^2)$ and intersecting it with $\text{Bir}(\mathbb{P}_\mathbb{R}^2)$. But there is no guarantee that $\text{Bir}(\mathbb{P}_\mathbb{R}^2)\cap N$ is non-trivial. How does one choose an $N$ s.t $\text{Bir}(\mathbb{P}_\mathbb{R}^2)\cap N$ is nontrivial?