A Belavin-Drinfeld triple associated to a simple Lie algebra $L$ is a triple $(\Gamma_1, \Gamma_2, \tau)$ where $\Gamma_1, \Gamma_2 \subseteq \Gamma$ ($\Gamma$ is a set of simple roots or fundamental roots of $L$) and $\tau : \Gamma_1 \longrightarrow \Gamma_2$ is a bijection such that

$(1)$ $\tau$ preserves the dual Killing form $(\cdot, \cdot),$

$(2)$ for any $\gamma_1 \in \Gamma_1,$ there exists $n \gt 0$ such that $\tau^n (\gamma_1) \in \Gamma_2 \setminus \Gamma_1.$

This definition has been taken from the lecture notes on compact quantum groups by Pavel Etingof and Oliver Schiffmann. In this connection the author mentioned that we can linearly extend $\tau$ as a map from $\mathbb Z \Gamma_1 \to \mathbb Z \Gamma_2.$ Let $L_{\Gamma_i}$ be the subalgebra generated by the $L_{\pm \alpha}, \alpha \in \Gamma_i$ for $i=1,2$ (i.e. $L_{\Gamma_i}$ is the direct sum of the root subspaces corresponding to the roots generated by the roots of $\Gamma_i$ for $i=1,2$). In other words, if $\Phi$ is the set of all roots of $L$ then $$L_{\Gamma_i} = \bigoplus_{\beta \in \mathbb Z \Gamma_i \cap \Phi} L_{\beta}$$ for $i=1,2.$ At this point the authors mentioned that the extended linear map $\tau$ from $\mathbb Z \Gamma_1 \longrightarrow \mathbb Z \Gamma_2$ would then induce an isomorphism from $L_{\Gamma_1} \longrightarrow L_{\Gamma_2}$ given by $e_{\beta} \mapsto e_{\tau(\beta)}$ and $f_{\beta} \mapsto f_{\tau(\beta)}$ where $L_{\beta} = \text {span}\ \{e_{\beta} \}$ and $L_{-\beta} = \text {span}\ \{f_{\beta}\},$ for $\beta \in \mathbb Z \Gamma_1 \cap \Phi$ which are known as Chevalley-Serre generators for the root subspaces of $L.$ This is where I am confused. I am unable to understand why is it even injective.

Question$:$ If $\beta \in \mathbb Z \Gamma_1 \cap \Phi$ is it always true that $\tau (\beta) \in \mathbb Z \Gamma_2 \cap \Phi\ $?

For otherwise injectivity of this induced map would be violated. Could anyone please shed some light on it?

Thanks for your time.

**Source** $:$ Lecture $5$ on Belavin-Drinfeld classification of Lie bialgebra structures on a complex simple Lie algebra from the *Lecture Notes on Compact Quantum Groups* written by Pavel Etingof and Oliver Schiffmann.