Is the adjoint map $G^0\to GL(\mathfrak g)$ proper for semi-simple $\mathfrak g$?

Question

Suppose that $G^0$ is the connected component of a semi-simple Lie group.

Is the adjoint representation $G^0\to GL(\mathfrak g)$ a proper map?

I'm not sure if this is an obvious statement or not. I definitely don't have any proof ideas about this (for simple examples like $SL_2$ it seems to be true).