For a hermitian symmetric space $M$ one has its group of biholomorphic maps $\operatorname{Hol}(M)$ and its group of Riemannian isometries $\operatorname{Isom}(M)$. According to Prop. 1.6 of [Milne - Introduction to Shimura varieties](https://www.jmilne.org/math/xnotes/svi.pdf), the inclusion of their intersection into either of them yields an isomorphism of identity components.


Looking at $S^2=\operatorname P^1\mathbb C$ this seems to mean that $\operatorname{PSL}(2,\mathbb C)$ is isomorphic to $\operatorname{SO}(3)$, which is of course wrong. So what am I misunderstanding here?



Automorphism group of Hermitian symmetric spaces
To get this question off the unanswered list:


In Proposition 1.6 Milne assumes that $X$ is a **Hermitian symmetric domain**, equivalently, a Hermitian symmetric space of noncompact type. This assumption rules out examples such as complex-projective spaces and complex-affine spaces as well as spaces containing such direct factors (for all of these the conclusion of the proposition fails).


