  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
$\mbox{O}(n)$.
