I am looking for a reference containing the following result:



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> Let $a$ and $b$ be two elements of a right-angled Artin group $A$. Assume that $a$ and $b$ have minimal length (with respect to the canonical generating set of $A$) in their conjugacy classes. Let $a\_1 \cdots a\_n$ and $b\_1 \cdots b\_m$ be words of minimal length representing $a$ and $b$ respectively. If $a$ and $b$ are conjugate in $A$, then $a\_1 \cdots a\_n$ can be obtained from $b\_1 \cdots b\_m$ by applying the following operations: permutation of two successive letters which commute, and cyclic permutation.
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I am sure that it is written somewhere, but I am not able to find where.



Conjugacy in right-angled Artin groups
Look at Lemma 9 of <https://arxiv.org/abs/0802.1771> for what you want. 


