Let $\Phi$ be a root system of a finite reflection group $W$. Let $\Delta$ be a simple system in $\Phi$. I want to prove that $\Delta$ is "the smallest" set which generates $W$, more precisely: There is no proper subset $A$ of $\Delta$ which generates $W$.
I start with the following: suppose well, then we can find $\alpha\in\Delta\setminus A$ such that $s_{\alpha}$ is not needed as a generator. But now? Is the goal to prove that $A=\Delta$ or how to get the contradiction?
Thank you for help.
Assume $\Delta$ irreducible, otherwise take components. The elements of $\Delta$ that are combinations of basic roots from $A$ form a root system $\Phi'\subset \Phi$. The reflection $(s_{\alpha})_{\alpha \in A}$ generate its Weyl group $W'$. Let $\beta \ne \alpha$, $\beta \in \Delta'$. We have $W' \cdot \beta \subset \Phi'$. Note that $\Phi' \ne \Phi$ since $\alpha$ is not an integral combination of the other basic roots. However, $W\cdot \beta = \Phi$ since $\Delta$ is irreducible. We conclude that $W'\ne W$.