We know that $so(3)$ has the explicit quadratic Casimir
$$L^2=\sum L_{i}^2.$$
Are there analogs to this in other simple lie algebras?
I know that for a simple lie algebra I can always use the Cartan metric to obtain a formula for the quadratic Casimir; $$C=g^{ij}v_{i}v_{j}.$$
But then there is the problem of actually finding the Cartan metric, which (correct me if i'm wrong) can be troublesome if your algebra is high dimensional.
In particular, i'm interested in the quadratic Casimir of the simple Lie algebra $sp(2N)$. If anyone has a link where this Casimir is given explicitly (as is the Casimir for $so(3)$) I would appreciate it very much. Or even some insight into how one would get an explicit form for the cartan metric would be much obliged.
