Let $G$ be a compact Lie group with Haar measure $dg$ and (finite-dimensional real) Lie algebra $\frak g$. Endow $\frak g$ with an $\hbox{Ad}$-invariant norm $\|\cdot\|_{\frak g}$ so that $\frak g$ becomes a normed space (one can use the norm induced by the negative of the Killing form).

Define the function $$ f:{\frak g}\to{\frak g},~~x\mapsto\int_Ggxg^{-1}dg, $$ where the integral is the Bochner integral associated with the norm $\|\cdot\|_{\frak g}$.

$f(x)$ exists for each $x\in\frak g$ because the norm of the integrand is bounded: $$ \|gxg^{-1}\|_{\frak g}=\|\hbox{Ad}_gx\|_{\frak g}=\|x\|_{\frak g}. $$ $f$ is $\hbox{Ad}$-invariant because $f(gxg^{-1})=f(x)$ for all $x\in\frak g$ and $g\in G$. Finally, some direct calculations using explicit parametrizations of $G$ and $\frak g$ show that $f$ is identically zero for certain groups $G$ and nonzero for others.

My questions are thus the following:

1) Can we find a general class of compact Lie groups $G$ such that $f$ is identically zero, or nonzero?

2) Is $f$ identically zero for all semisimple compact Lie groups G?

(maybe some Weyl integration formula or some argument based on Casimir functions can be used, but I have not been able to do it...)

Thank you for the help!