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$\DeclareMathOperator\SO{SO}$Fix even $n=2m\ge 4$. Let $G_0$ be the copy of $\SO(m)\times\SO(m)$ in $\SO(2m)$. Let $G$ be generated by $G_0$ and an element of order 2 switching the two copies of $\mat …

Yes, here'a a proof by induction, granted the $n=2$ case (which is the only one where [basic] arithmetic occurs). Let $G$ be the closure. I first claim that $G$ acts transitively on the sphere. Indeed …

Let $n=m^2$ be a square. Then the vector $(1,\dots,1)\in\mathbf{Q}^n$ has norm $m$, as does $(0,\dots 0,m)$. Hence by Witt's theorem there exists an element $u$ of $\mathrm{O}_n(\mathbf{Q})$ mapping $ …

No. Indeed $\mathrm{SO}(4)$ satisfies the following condition (which is a first-order existential formula) but not $\mathrm{SU}(3)$: $$\exists w,x,y,z: [x,w]\neq 1\neq [y,z],\; [w,y]=[w,z]=[x,y]=[x,z] …

Here's a proof assuming $n\ge 3,n\neq 8$ that the Lie algebras are isomorphic only when the quadratic forms are equivalent up to rescaling (I assume $K$ has characteristic zero and fix an algebraicall …