Direct Product of Weyl Groups

Question

We define the Weyl group $W(R)$ of a root system $R$ as the group generated by the reflections $s_{\alpha}$ for $\alpha$ $\in$ $R$. Then show that $W(R)$ is isomorphic to the direct product of the respective Weyl groups of its irreducible components.There are proofs available of this statement in this site by using the alternative Lie group theoretic definition of $W(R)$. But I am trying to prove this statement by using this definition and related results in the theory of abstract root systems on a Euclidean space. Can anybody please help me with this problem? Thanks for any help.

Answer 1

Assume $R=R_1\sqcup R_2$ with $\alpha\perp\beta$ for all $\alpha\in R_1$ and $\beta\in R_2$. We regard $W(R_1), W(R_2)\leq W(R)$ in the obvious way.

Then, it is easy to see that $s_\alpha s_\beta=s_\beta s_\alpha$ for all $\alpha\in R_1$ and $\beta\in R_2$ since $\alpha\in\bigcap_{\beta\in R_2}\ker s_\beta$ (and vice-versa). Given this fact, it is also easy to see that $W(R_1)W(R_2)=W(R)$ and $W(R_1)\cap W(R_2)=\{1\}$ (since $R_1\sqcup R_2=R$ and $R_1\cap R_2=\emptyset$). It is now a basic group theory result that $$W(R)\cong W(R_1)\times W(R_2)$$