The weyl group as a dihedral group

Question

I am traying to solve problem 9.4 of humphreys lie algebra book and I need to show that the Weyl groups of $A_1 \times A_1$ $A_2$ $B_2$ and $G_2$ are dihedral of order 4,6,8,12.

My question is, how can a reflection group be a dihedral group since every element in a reflection group is of order two?

Answer 1

For $A_1$X $A_1 $, consider the Weyl group: $\{Id, \sigma_\alpha, \sigma_\beta, \sigma_\alpha \sigma_\beta\}$.($ Id $ is identity map)

It's group of order 4. It satisfies the properties of Dihedral group, you can verify it!