In [the paper](http://www.ams.org/journals/proc/1994-120-03/S0002-9939-1994-1169886-7/S0002-9939-1994-1169886-7.pdf), a set $L$ associated to an element $w$ in a Coxeter group $W$ is defined as follows. Let $w=s\_{i\_1} \cdots s\_{i\_m}$ be a reduced expression. Define $L=\{\beta\_1, \ldots, \beta\_m\}$, where
\begin{align}
& \beta\_1 = \alpha\_{i\_1}, \\
& \beta\_k = s\_{i\_1} \cdots s\_{i\_{k-1}}(\alpha\_{i\_k}), k=2,3,\ldots,m.
\end{align}
The set $L$ is denoted by $R\_w$. If $w$ is reduced, $\beta\_1, \ldots, \beta\_m$ are different from each other. I think that the following is true. If $w=s\_{i\_1} \cdots s\_{i\_m}$ is not reduced, then there are two elements $\alpha, -\alpha$ in $\beta\_1, \ldots, \beta\_m$ whose sum is $0$. Is this true? Are there some references about this?


In the case of type $A\_2$, for $w= s\_1 s\_2 s\_1 s\_2$ which is not reduced, we have
\begin{align}
\beta\_1 = \alpha\_1, \beta\_2 = \alpha\_1+\alpha\_2, \beta\_3 = \alpha\_2, \beta\_4=-\alpha\_1.
\end{align}
We have $\alpha\_1+(-\alpha\_1)=0$. Thank you very much.



Positive roots and elements in a Coxeter group.
This is a standard observation in Lie theory: if you let the Weyl group act on the root system on the left, $\beta\_k$ is the unique root such that $s\_{i\_k}\cdots s\_{i\_1}\beta\_k$ is negative and $s\_{i\_{k-1}}\cdots s\_{i\_1}\beta\_k$ is positive. That is, for a positive root $\gamma$, the number of times it appears as $\pm \beta\_k$ is the number of times that $s\_{i\_k}\cdots s\_{i\_1}\gamma$ changes sign as $k$ goes from 0 to $m$, and it's $\beta\_k$ if the change is positive to negative, and $-\beta\_k$ if the change is negative to positive. 


An expression is unreduced if and only if there is a positive root that changes sign twice, so the places where this sign change happens gives the cancelling $\beta$'s. 


