I have a question concerning centralizers in real reductive groups. I'd like to know if the following property is available in any references.

Let $L\subset H\subset G$ be an inclusion chain of connected semi-simple algebraic groups defined over the real numbers $\mathbb{R}$. Assume that the centralizers $Z(L,H)$ and $Z(H,G)$ are both compact, can one show that $Z(L,G)$ is also compact? Here one may add the condition that $L$, $H$ are bth stable under a Cartan involution $\theta$ of $G$.

Or more algebraically, if the groups above are defined over a perfect field $k$ (say of characteristic zero), and that $Z(L,H)$ and $Z(H,G)$ are both $k$-anisotropic, in the sense that they do not contain split $k$-torus of dimension $>0$, then can one show that $Z(L,G)$ is also $k$-anisotropic?

Any examples and counter-examples are also welcome.

Many thanks