Let $A=(a_{ij})_{1 \leq i,j \leq n}$ be an indecomposable symmetrizable generalized Cartan matrix. Then $A$ is the Cartan matrix of a finite reduced crystallographic root system if and only if there exists a diagonal matrix $D=\operatorname{diag}(d_1,\dots,d_n)$ with $d_i>0$ for every $1 \leq i \leq n$ such that $DA$ is symmetric positive definite.