Simply put, how do I calculate (in general) the central elements of the UEA of some Lie algebra given some desired degree in the algebra generators?
I know the so-called 'quadratic Casimir', of degree $2$, is $$ X_i X^i = \kappa_{ij}X^i X^j $$ for the Killing form $\kappa_{ij}$. This is much faster than, say, writing out some arbitrary $A_{ij} X^i X^j$ and repeatedly applying the commutation relations until there are enough constraints on the $A_{ij}$ to create a central element.
But what about if I want some other central elements? For instance the 'cubic Casimir' might be an appropriate name for the central element of degree $3$.
I don't want to have to begin with a general $B_{ijk} X^i X^j X^k$ and compute from there, as this contains a potentially enormous number of third degree monomials. Even in the most elementary case of $\mathfrak{sl}_3$, this is extremely daunting.
Is there a faster way?