Let $P$ be the root poset associated to a simple Lie algebra.
Let $L=L(P)$ denote the distributive lattice of order ideals of $P$ and let $Co_L$ denote the Coxeter matrix of $L(P)$ which is defined as $-C^{-1} C^T$ when $C$ is the matrix with entries $c_{i,j}=1$ when $i \leq j$ and $c_{i,j}=0$ else for $i,j \in L$.

>Question: For which types is the matrix $Co_L$ periodic? That is when do we have $Co_L^t=id$ for some $t \geq 1$? What is the period (the smallest such $t$) in case this is true for a given type?

The quesiton has a positive answer for all simple Lie algebras of rank at most 5. So maybe it is always periodic? For example in type $A_5$ the period is 14 and for $D_5$ it is 18.