Automorphism group of Hermitian symmetric spaces

Question

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, the inclusion of their intersection into either of them yields an isomorphism of identity components.
Looking at $S^2=operatorname P^1mathbb C$ 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?

Moishe Kohan · Accepted Answer

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).

Automorphism group of Hermitian symmetric spaces

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