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Let $g$ be a simple Lie algebra over an algebraically closed field ofcharacteristic zero. The goal of this note is to prove a closed formula for theDynkin index of a principal $sl_2$-subalgebra of $g$. The key step in the proofuses the "strange formula" of Freudenthal--de Vries. As an application, we (1)compute the Dynkin index any simple $g$-module regarded as $sl_2$-module and(2) obtain an identity connecting the exponents of $g$ and the dual Coxeternumbers of both $g$ and the Langlands dual $g^\vee$.

On the Dynkin index of a principal $\mathfrak{sl}_2$-subalgebra