can be raised and lowered freely. We denote by ${\rm dim}(R)$ the dimension of $R$, and by $C(R)$ the value of the quadratic Casimir operator in the representation $R$, [*i.e.*]{}  $\sum_a t^a t^a = C(R) \times{\rm identity}$. In these conventions $${\rm Tr}_R\;(t^a t^b) = I_R \, \delta^{ab}\ , \ \ \   {\rm where}\
\ \ \ I_R = {C(R)\times  {\rm dim}(R)  \over {\rm dim}(g)}\ ,$$ where ${\rm dim}(g)$ is the dimension of the adjoint representation. Furthermore, the dual Coxeter number $h^\vee$ is given by : $$I_{adj}\;\delta^{ab} = \sum_{a,b} f^{abc} f^{abd}
= h^\vee\psi^2 \delta^{ab}\,,$$ where $\psi^2$ is the length squared of the longest root. We will also need the trace of triple products of generators, $$\begin{aligned}
{\rm Tr}_R \; (t^a t^b t^c) & = &
{1\over 2}\; {\rm Tr}_R([t^a,t^b] t^c)
+ {1\over 2}\; {\rm Tr}_R(\{t^a, t^b\} t^c)  \nonumber \\
& = & {i\over 2}  \, f^{abc}\, I_R
+ {1\over 2} \, d^{abc} \, I_R^{(3)}\, ,\end{aligned}$$ where $d^{abc}$ is the totally symmetric invariant third order tensor, and $I_R^{(3)}$ is related to the value in the representation $R$ of the associated third order Casimir operator, $$C_3 = \sum_{abc} d^{abc}\, t^a \, t^b\, t^c\ .$$ Note that the $d^{abc}$ may vanish, as happens for 