This should be an easy question about centralizers in reductive lie groups, but I wonder if it is already available from the literature.
Consider $G$ a connected non-compact semi-simple Lie group, with a Cartan involution $\sigma$, and $H$ a reductive subgroup, stablized by $\sigma$. Then the fixed part of $\sigma$ in $H$ is a maximal compact subgroup $K_H$ in $H$. Compare the two centralizers $Z(K_H,G)$ and $Z(H,G)$. Are they equal, or at least, their derived parts equal to each other?