Let $A = (a_{ij})_{1 \leq i,j \leq n}$ and $A' = (a'_{ij})_{1 \leq i,j \leq n}$ be finite-type Cartan matrices. Let $\Gamma(A)$ and $\Gamma(A')$ be their Dynkin diagrams, with one vertex for each index, with $a_{ij}a_{ji}$ edges between distinct vertices $i$ and $j$ when $a_{ij}a_{ji} \neq 0$, and with the arrow, when present, directed toward the shorter simple root equivalently toward the vertex $i$ for which $|a_{ij}| > |a_{ji}|$.

If $\Gamma(A)$ and $\Gamma(A')$ are isomorphic as Dynkin diagrams, then there exists a permutation $\sigma$ of $\{1,\dots,n\}$ such that

\begin{align*} a'_{\sigma(i)\sigma(j)} = a_{ij} \end{align*}

for all $1 \leq i,j \leq n$. Equivalently, the Dynkin diagram determines the Cartan matrix up to simultaneous permutation of rows and columns.