The Boolean SATisfiability Problem and the orthogonal group $O(n)$

We explore the relations between the Boolean Satisfiability Problem with $n$ Boolean variables and the orthogonal group $\mbox{O}(n)$. We show that all $2^n$ possible solutions induce involutions of $\mathbb{R}^n$ that lie in the compact, disconnected real manifold of dimension $n (n-1)/2$ of $\mbox{O}(n)$. This result in turn gives a new unsatisfiability test within group $\mbox{O}(n)$.