# Killing metric on compact simple groups

1. Jul 29, 2014

### center o bass

The killing form can be defined as the two-form $K(X,Y) = \text{Tr} ad(X) \circ ad(Y)$ and it has matrix components $K_{ab} = c^{c}_{\ \ ad} c^{d}_{\ \ bc}$. I have often seen it stated that for a compact simple group we can normalize this metric by
$K_{ab} = k \delta_{ab}$ for some proportionality constant $k$. How is this statement proved?

Last edited: Jul 29, 2014
2. Aug 14, 2014