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[upd: this answers the old version of the question, which has since been changed.] The intersection is $SO(2)^n=SO(2)\times SO(2)\times\cdots\times SO(2)$, as perhaps expected. The inclusion $SO(2)^n\subset (SL_2)^n\cap SO(2n)$ is clear. The other inclusion follows from this observation: If a linear map that preserves a vector subspace is to be orthogonal, it must preserve the metric restricted to the subspace. |
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The intersection is $SO(2)^n=SO(2)\times SO(2)\times\cdots\times SO(2)$, as perhaps expected. The inclusion $SO(2)^n\subset (SL_2)^n\cap SO(2n)$ is clear. The other inclusion follows from this observation: If a linear map that preserves a vector subspace is to be orthogonal, it must preserve the metric restricted to the subspace. |
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