Let $W$ be a Weyl group of a semisimple Lie algebra $\mathfrak{g}$. Let $\beta$ be a positive root, $\alpha$ a simple root such that $\beta=w(\alpha)$.
It is a straightforward fact that we can express the reflection through $\beta$ as $$s_\beta=ws_\alpha w^{-1}$$
Question: Does this give a reduced expression for $s_\beta$? If not, is there a nice way to construct reduced expressions for arbitrary reflections in $W$?
This doesn't always give a reduced expression: for example, in Type $A_2$ writing $s_1$ and $s_2$ for the simple reflections, we have $s_1 = (s_1s_2s_1)s_2(s_1s_2s_1)$, which is not a reduced expression.
But given $\beta$, one can make a choice of $\alpha$ and $w$ such that $s_\beta = ws_\alpha w^{-1}$ is a reduced expression. This is explained in Lemma 2.7 of "Reflection Subgroups of Coxeter Systems" by Matthew Dyer. It is also known that every reduced word for any reflection arises in this way; this is shown in Proposition 2.4 of "Quasiminiscule Quotients and Reduced Words for Reflections" by John Stembridge.