The dual Coxeter number 
of a complex simple Lie algebra
is the normalization constant relating its Killing form 
to the invariant bilinear form 
normalized so that long roots have squared
length 2,
 |
For 
with 
,
(Fox 2026). The reciprocal 
is the optimal constant in the Lie algebra extension
of the Böttcher-Wenzel inequality
for the Hermitian norm induced by a compact real form (Fox 2026).
