Casimir element

Let $\mathfrak{g}$ have a nondegenerate symmetric invariant bilinear form on a Lie algebra $B$, and choose a basis $(x_i)$ with $B$-dual basis $(x^i)$. Its quadratic Casimir element isThe inverse-form tensor ${\sum }_i{x}_i\otimes x^i$ is independent of the basis and invariant under the Adjoint representation of a Lie algebra. Multiplying its factors in the universal enveloping algebra gives $[z,C_B]={\sum }_i([z,x_i]x^i+x_i[z,x^i])=0$, so $C_B$ is central. In a finite-dimensional irreducible complex representation, the Schur lemma makes its represented quadratic Casimir operator a scalar.^[ref]

With the Killing form on a complex semisimple Lie algebra, its Casimir eigenvalue on highest weight $\lambda$ is $(\lambda ,\lambda +2\rho )$, with the induced dual inner product and Weyl vector $\rho$. Scaling $B$ by $c\ne 0$ scales $C_B$ by $c^{-1}$. For the sl2 Lie algebra, the normalization $B(e,f)=1$, $B(h,h)=2$ gives $ef+fe+h^2/2$; the actual Killing form is four times this form, so its Casimir is one quarter of that expression.