Let $\mathfrak{g}$ be a semi-simple Lie algebra and let $\phi:\mathfrak{g}\rightarrow\mathfrak{gl}(V)$ be its finite-dimensional complex irreducible representation. You can define two non-degenerate symmetric forms on $\mathfrak{g}$:

Standard Killing form: $K(X,Y)=tr(ad_X\circ ad_Y)$

"Killing-like" form associated with $\phi$: $K_{\phi}(X,Y)= tr(\phi(X)\phi(Y))$

In general, is there any connection between $K_\phi$ and $K$? I know for instance that for defining representation of $\mathfrak{gl}(N,\mathbb{C})$ both forms are proportional. Is it true for general semi-simple Lie algebra? If not, is there a separate name for $K_{\phi}$?

I'm asking this question because in math-physical literature connected to research I'm doing people tend to confuse these two forms: $K_\phi$ is used to define second order Casimir invariant of a given representation. Yet, in some articles there is simply $K$ instead of $K_{\phi}$.

`$\mathbb{C}$`

. Here a "Casimir element" attached to a representation and trace function is contrasted with a "universal Casimir element" defined using the Killing form. To treat semisimple (or reductive) rather than just simple Lie algebras, you just need to be careful about the ideals acting as zero in a representation. $\endgroup$ – Jim Humphreys Dec 17 '10 at 14:34