Weyl groups: correspondence of reflections and roots?

Question

If W is the Weyl group of some ADE-type Lie algebra, and w is an element corresponding to a reflection (not just an involution), does it necessarily correspond to a root?

Answer 1

Choose a base of simple roots $x_1,x_2,\ldots,x_n$ corresponding to simple reflections $s_1,s_2,\ldots,s_n$. If $t$ is a reflection, then there exists a $w$ in the Weyl group and an $i$ such that $t=ws_iw^{-1}$ . Then the root corresponding to this reflection is $w(x_i)$. This construction may give either a positive or a negative root, but if $w$ is if minimal length then the root will be positive. The correspondence goes both ways, as there is a bijection between reflections and positive roots.