Let $A=(a_{ij})_{1 \le i,j \le n}$ be a finite-type generalized Cartan matrix, and let $\Gamma(A)$ be its Dynkin diagram with vertex set $I=\{1,\dots,n\}$. Let
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\begin{align*}
I = I_1 \sqcup \cdots \sqcup I_r
\end{align*}
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be the partition of $I$ into the connected components of $\Gamma(A)$, and let $\Gamma_k$ denote the full connected subdiagram on $I_k$. Then there exists a permutation $\sigma$ of $I$ such that the simultaneously permuted matrix