Strange identity of Freudenthal-de Vries

introduction

Root Systems and Dynkin diagrams
\(\rho\) Weyl vector
Kac book 219p, 221p
strange formula \[\frac{\langle\rho,\rho\rangle}{2h^{\vee}}=\frac{\operatorname{dim}\mathfrak{g}}{24}\]
very strange formula
conformal anomaly \[m_{\Lambda}=\frac{(\Lambda+\rho)^2}{2(k+h^{\vee})}-\frac{\rho^2}{2h^{\vee}}=h_{\lambda}-\frac{c(k)}{24}\]

Thiel, Marko, and Nathan Williams. “Strange Expectations.” arXiv:1508.05293 [math], August 21, 2015. http://arxiv.org/abs/1508.05293.
H. FREUDENTHAL and H. DE VRIES. “Linear Lie groups”, New York: Academic Press, 1969.
AN ELEMENTARY PROOF OF THE 'STRANGE FORMULA' OF FREUDENTHAL AND DE Vries
- John Burn, 2004