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Mikhail Borovoi
Mikhail Borovoi
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YCor
YCor
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Let $\mathbf{G}$ be the image of the natural embedding of $\operatorname{SL}_4(\mathbb{R})$ inside $\operatorname{SL}_4(\mathbb{C})\subset \operatorname{SL}_8(\mathbb{R})$. Then $\mathbf{G}$ is an algebraic group defined over $\mathbb{Q}$ with $\mathbf{G}(\mathbb{Q})$ being $\mathbb{Q}$-isotropic.

My question: is it possible to find $g\in \operatorname{SL}_8(\mathbb{R})$ such that $g\mathbf{G}g^{-1}$ is $\mathbb{Q}$-algebraic and $\mathbb{Q}$-$\mathbf{anisotropic}$anisotropic?

Let $\mathbf{G}$ be the image of the natural embedding of $\operatorname{SL}_4(\mathbb{R})$ inside $\operatorname{SL}_4(\mathbb{C})\subset \operatorname{SL}_8(\mathbb{R})$. Then $\mathbf{G}$ is an algebraic group defined over $\mathbb{Q}$ with $\mathbf{G}(\mathbb{Q})$ being $\mathbb{Q}$-isotropic.

My question: is it possible to find $g\in \operatorname{SL}_8(\mathbb{R})$ such that $g\mathbf{G}g^{-1}$ is $\mathbb{Q}$-algebraic and $\mathbb{Q}$-$\mathbf{anisotropic}$?

Let $\mathbf{G}$ be the image of the natural embedding of $\operatorname{SL}_4(\mathbb{R})$ inside $\operatorname{SL}_4(\mathbb{C})\subset \operatorname{SL}_8(\mathbb{R})$. Then $\mathbf{G}$ is an algebraic group defined over $\mathbb{Q}$ with $\mathbf{G}(\mathbb{Q})$ being $\mathbb{Q}$-isotropic.

My question: is it possible to find $g\in \operatorname{SL}_8(\mathbb{R})$ such that $g\mathbf{G}g^{-1}$ is $\mathbb{Q}$-algebraic and $\mathbb{Q}$-anisotropic?

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user515519
user515519

$\mathbb{Q}$-forms of $\operatorname{SL}_4(\mathbb{R})$ inside $\operatorname{SL}_8(\mathbb{R})$

Let $\mathbf{G}$ be the image of the natural embedding of $\operatorname{SL}_4(\mathbb{R})$ inside $\operatorname{SL}_4(\mathbb{C})\subset \operatorname{SL}_8(\mathbb{R})$. Then $\mathbf{G}$ is an algebraic group defined over $\mathbb{Q}$ with $\mathbf{G}(\mathbb{Q})$ being $\mathbb{Q}$-isotropic.

My question: is it possible to find $g\in \operatorname{SL}_8(\mathbb{R})$ such that $g\mathbf{G}g^{-1}$ is $\mathbb{Q}$-algebraic and $\mathbb{Q}$-$\mathbf{anisotropic}$?