Dynkin index

In mathematics, the Dynkin index 
of a representation with highest weight formula_2 of a compact simple Lie algebra formula_3 that has a highest weight formula_4 is defined by
evaluated in the representation formula_2. Here formula_7 are the matrices representing the generators, and 
formula_8 is given by
evaluated in the defining representation.
By taking traces, we find that 
where the Weyl vector
is equal to half of the sum of all the positive roots of formula_3. The expression formula_13 is the value of the quadratic Casimir in the representation formula_2. The index formula_1 is always a positive integer.
In the particular case where formula_4 is the highest root, meaning that formula_2 is the adjoint representation, formula_1 is equal to the dual Coxeter number.