(This is a follow-up to [this question][1] of mine.)

Is there an example of a connected reductive algebraic group $G$ over $\mathbb{R}$ such that:

- $G$ is not isomorphic to a product $G_1 \times G_2$ of smaller groups (*isogenous* to a product is OK)
- $G$ is not a torus,
- the quotient of $G$ by a maximal compact-mod-centre subgroup has a complex structure,
- $Z_G(\mathbb{R})$ is not contained in the identity component of $G(\mathbb{R})$?


The condition $Z_G(\mathbb{R}) \subseteq G(\mathbb{R})^\circ$ is vacuously satisfied if $G$ is adjoint, because then $Z_G = \{1\}$; but it is also vacuously satisfied if $G$ is semisimple and simply-connected, because then $G(\mathbb{R})$ is connected as a Lie group by a theorem of Cartan. So any example would have to lie somewhere in between the two (which makes me wonder if there are any examples at all).

PS: Of course $GL_3$ is an example if the "complex structure" condition is dropped. 

  [1]: https://mathoverflow.net/questions/333846/shimura-varieties-and-connected-components