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