We assume $\mathbb{K} = \mathbb{R}$. We denote by $O _ { r , s } : = j \left( O \left( Q _ { r , s } \right) \right)$ the subset of matrices associated to the orthogonal group $O \left( Q _ { r , s } \right)$ of $Q _ { r , s }$. Prove that $O _ { r , s }$ is a closed subgroup of $\mathrm { GL } _ { n } ( \mathbb { R } )$ (we equip $\mathcal { M } _ { n } ( \mathbb { R } )$, the set of square matrices of size $n$ with coefficients in $\mathbb { R }$, with its topology as a $\mathbb { R }$-vector space of finite dimension).
We assume $\mathbb{K} = \mathbb{R}$. We denote by $O _ { r , s } : = j \left( O \left( Q _ { r , s } \right) \right)$ the subset of matrices associated to the orthogonal group $O \left( Q _ { r , s } \right)$ of $Q _ { r , s }$. Prove that $O _ { r , s }$ is a closed subgroup of $\mathrm { GL } _ { n } ( \mathbb { R } )$ (we equip $\mathcal { M } _ { n } ( \mathbb { R } )$, the set of square matrices of size $n$ with coefficients in $\mathbb { R }$, with its topology as a $\mathbb { R }$-vector space of finite dimension).