Does there exists a reductive group $G$ of type $E_7$ over a given totally real field $L^+$ such that for every embedding $\tau:L^+\to \mathbb{R}$ except one , $G_{\tau,\mathbb{R}}(\mathbb{R})$ is compact? Is there a way to construct a group with a given number of compact form?

Does there exist a reductive group $G$ of type $E_7$ over a given totally real field $L^+$ such that for every embedding $\tau:L^+\to \mathbb{R}$ except one , $G_{\tau,\mathbb{R}}(\mathbb{R})$ is compact? Is there a way to construct a group with a given number of compact forms?

Does there exists a reductive group $G$ of type $E_7$ over a given totally real field $\mathbb{R}$ such that for every embedding $\tau:L^+\to \mathbb{R}$ expect one , $G_{\tau,\mathbb{R}}(\mathbb{R})$ is compact? Is there a way to construct a group with a given number of compact form?

Does there exists a reductive group $G$ of type $E_7$ over a given totally real field $L^+$ such that for every embedding $\tau:L^+\to \mathbb{R}$ except one , $G_{\tau,\mathbb{R}}(\mathbb{R})$ is compact? Is there a way to construct a group with a given number of compact form?