For a simple Lie group $G$ and representation $\rho:G\to GL(n,\mathbb C)$ with 
infinitesimal representation $\rho':\mathfrak g \to \mathfrak g\mathfrak l(n,\mathbb C)$ we have 
$Trace(\rho'(X).\rho'(Y)) = j_{\rho} B(X,Y)$ where $B$ is the Cartan-Killing form, for a constant $j_{\rho}$, which is called the Dynkin index (up to a possible factor due to normalization). This is, because the quadratic bi-invariant functions form a 1-dim. vector space in this case. 

So it is already isometric up to a conformal factor. Multiply it away. 
You do this for each simple part of the Lie algebra, stacking matrices, and get it for semisimple compact Lie groups. 
Then you play with the center.