Let $k$ be an algebraically closed field of characteristic $0$. Let $L$ and $M$ be finite-dimensional semisimple Lie algebras over $k$. Choose Cartan subalgebras $H_L \subset L$ and $H_M \subset M$, and choose ordered bases of simple roots $\Delta_L = (\alpha_1,\dots,\alpha_\ell)$ and $\Delta_M = (\beta_1,\dots,\beta_\ell)$ for their respective root systems. Let $A_L = (a_{ij})_{1 \leq i,j \leq \ell}$ and $A_M = (b_{ij})_{1 \leq i,j \leq \ell}$ be the corresponding Cartan matrices, where

\begin{align*} a_{ij} = \alpha_j(h_i), \qquad b_{ij} = \beta_j(h_i^M), \end{align*}

with $h_i \in H_L$ and $h_i^M \in H_M$ the coroot elements determined by the chosen simple roots.

If there exists a permutation $\pi$ of $\{1,\dots,\ell\}$ such that

\begin{align*} b_{ij} = a_{\pi(i)\pi(j)} \end{align*}

for all $1 \leq i,j \leq \ell$, then $L$ and $M$ are isomorphic as Lie algebras over $k$.