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 invariant symmetric forms on $\mathfrak{g}$:

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

2. "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 $K_{\phi}$.

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 $\mathfrak{g}$:

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

2. "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 $K_{\phi}$.