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 $O(n)$. We show that all $2^n$ possible solutions induce involutions of ${\mathbb{R}}^n$ that lie in the compact, disconnected real manifold of dimension $n(n-1)\mathrm{/}2$ of $O(n)$. This result in turn gives a new unsatisfiability test within group $O(n)$.