In Margulis' book, there are actually two definitions of arithmetic lattices:
- If $\mathbf{G}$ is a connected semisimple algebraic $\mathbb{R}$-group, then a lattice $\Gamma \subset \mathbf{G}(\mathbb{R})$ is arithmetic if there exists a connected almost $\mathbb{Q}$-simple $\mathbb{Q}$-group $\mathbf{F}$ and an $\mathbb{R}$-epimorphism $\tau \colon \mathbf{F} \to \mathbf{G}$ such that $(\ker\tau)(\mathbb{R})$ is compact and $\tau(\mathbf{F}(\mathbb{Z}))$ is commensurable with $\Gamma$.
- If $H$ is a connected semisimple real Lie group, then a lattice $\Gamma \subset H$ is arithmetic if $\text{Ad}\,\Gamma$ is arithmetic in $(\overline{\text{Ad}\,H})(\mathbb{R})$ in the above sense.
As far as I understand it, the second definition is necessary because $H$ itself may not be of the form $\mathbf{G}(\mathbb{R})$. But if this is the case, i.e. if $H =\mathbf{G}(\mathbb{R})$ for some connected semisimple linear $\mathbb{R}$-group $\mathbf{G}$, is then the second definition equivalent to the first one?
Assuming the answer is yes, it would be interesting to know for which Lie groups $H$ the second definition is actually neccessary. If $H$ is not linear, then it clearly is. But if $H$ has finite center, can we always find a group $\mathbf{G}$ with $H = \mathbf{G}(\mathbb{R})$?
