Let $k$ be an algebraically closed field of characteristic $0$, and let $A=(a_{ij})_{1 \leq i,j \leq l}$ be a finite-type Cartan matrix. Let $\mathfrak{g}$ be the semisimple Lie algebra over $k$ associated to $A$, with Chevalley generators $e_i,f_i,h_i$ for $1 \leq i \leq l$.
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Then $\mathfrak{g}$ is isomorphic to the Lie algebra generated by elements $e_i,f_i,h_i$ for $1 \leq i \leq l$, subject exactly to the relations
for all distinct indices $i,j$ with $1 \leq i,j \leq l$.
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Equivalently, the canonical homomorphism from the Lie algebra defined by these generators and relations onto the semisimple Lie algebra with Cartan matrix $A$ is an isomorphism.