# Traces over the generators of a Lie algebra.

Let $\{t^a\}_{a\in[1,r]}$ generate a simple algebra $\mathfrak g$. Given a representation $R$ thereof, we let $$\text{tr}_{R}(A):=\text{tr}(R(A))\tag1$$ for any $A\in\mathfrak g$.

Consider two (faithful, finite-dimensional) representations $R_1$, $R_2$.

Is it true that $$\text{tr}_{R_1}(t^{a_1}t^{a_2}\cdots t^{a_n})\propto \text{tr}_{R_2}(t^{a_1}t^{a_2}\cdots t^{a_n})\tag2$$ ?

In the case of $n=0$, we have $$\frac{1}{\dim(R_1)}\text{tr}_{R_1}(\boldsymbol 1)=\frac{1}{\dim(R_2)}\text{tr}_{R_2}(\boldsymbol 1)\tag3$$ where both sides equal $1$.

In the case of $n=1$, both sides vanish.

In the case of $n=2$, we have $$\frac{1}{T(R_1)}\text{tr}_{R_1}(t^{a_1}t^{a_2})=\frac{1}{T(R_2)}\text{tr}_{R_2}(t^{a_1}t^{a_2})\tag4$$ where $T(R)$ is the so-called index of $R$. Here, both sides are in fact equal to $|\theta|^2\delta^{a_1a_2}$, with $\theta$ the highest root of $\mathfrak g$ (and this equation defines the index).

The case $n=3$ seems to be true, and the coefficient is the so-called anomaly coefficient, $A(R)$ (again, essentially defined by this equation; this is true at least in the case of $\mathfrak g=\mathfrak{su}(N)$; the anomaly coefficient is used in particle physics, but I cannot find a good reference right now).

Is the generalisation to arbitrary $n$ true? If so, what is the coefficient? Does it have a name? Can it be computed as a function of $\dim(R),T(R),A(R)$ only?