Eigenvalues of higher-order Casimirs of a Lie algebra Let $\mathfrak{g}$ be a simple Lie algebra of rank $r$. We work in the Cartan-Weyl basis with the Cartan subalgebra generated by $\{H_1,...,H_r\}$ and $\Phi$ the associated root system. In the Cartan-Weyl basis, The quadratic Casimir operator is given by
$$
C_2 = \sum_{i=1}^r H_iH_i + \sum_{\alpha\in\phi^+}(E_\alpha E_{-\alpha}+E_{-\alpha}E_{\alpha})
$$
with $\phi^+$ the positive roots and $E_\alpha$ the generator associated with the root $\alpha$.
Given a inner product $(\cdot,\cdot)$ over the root system and a representation over the finite-dimensional module $L(\Lambda)$ with highest weight $\Lambda$, the Casimir acts as
$$
C_2 L(\Lambda) = c_2 L(\Lambda)\,,\qquad c_2 = (\Lambda+ 2\rho,\Lambda)\,,
$$
where $\rho=\frac{1}{2}\sum_{\alpha\in\phi^+}\alpha$ the Weyl vector. This is explained in most textbooks, but it is usually the only Casimir discussed. Is there a way to find the $n$-th order Casimir value of a representation as a function of its highest weight? I.e. something of the form
$$
C_n L(\Lambda) = c_n L(\Lambda)\,,\qquad c_n = f(\Lambda)\,,
$$
I'm in particular interested in the quartic ($n=4$) Casimir eigenvalues.
 A: After looking at the literature, I have not found a closed formula that only involves the highest-weight state $\Lambda$ and simple quantities like the Weyl vector. What I was looking for is however related to the so-called $n$-th order representation indices introduced in [1]:
$$
 \ell_{2n} = \sum_{\mu\in W(\Lambda)}(\mu,\mu)^n
$$
where $W(\Lambda)$ is the weight system of the representation with highest-weight $\Lambda$. For $n=1$ this is the Dynkin index, which is (possibly up to normalisation) the eigenvalue of the quadratic Casimir described above.
Note that since there are only $r$ independent Casimir operators for an algebra of rank $r$, they are relations between them. There is a paper by Okubo [2] (and many follow up papers) showing how to find them.  They are also related to subalgebra embeddings $\mathfrak{h}\hookrightarrow\mathfrak{g}$  (embedding indices) and branching rules.
The indices are easily computed with e.g. Mathematica, but are also tabulated in [3].
Beware that there are different conventions, particularly between physicists and mathematicians.
[1] Patera, J.; Sharp, R. T.; Winternitz, P., Higher indices for group representations, J. Math. Phys. 17, 1972-1979 (1976). ZBL0347.22010.
[2] Okubo, Susumu, Modified fourth-order Casimir invariants and indices for simple Lie algebras, J. Math. Phys. 23, 8-20 (1982). ZBL0526.17001.
[3] McKay, W. G.; Patera, J., Tables of dimensions, indices, and branching rules for representations of simple Lie algebras, Lecture Notes in Pure and Applied Mathematics, Vol. 69. New York, Basel: Marcel Dekker, Inc. VII, 317 p. SFr. 90.00 (1981). ZBL0448.17001.
