Given a semi-simple Lie algebra $\mathfrak g$ the Killing form induces a canonical central element of $U(\mathfrak g)$. This is seen as a very important and useful thing.
Why is it useful to know what the Casimir element is?
I know one can use the existence of this Casimir element to show Weyl's theorem that representations of semi-simple Lie algebras are semi-simple, but what do I gain by knowing that, for example, the Casimir of $\mathfrak{so}_3$ is $L_1^2+L_2^2+L_3^2$?