A birational involution

Given a general K3 surface S of degree 18, lattice theoretic considerations allow to predict the existence of an anti-symplectic birational involution $\phi$ of the Hilbert cube $S^{[3]}$. We describe this involution in terms of the Mukai model of $S$, with the help of the famous transitive action of the exceptional group $G_2(R)$ on the six-dimensional sphere. We make a connection with Homological Projective Duality by showing that the indeterminacy locus of the involution is birational to a $P^2$-bundle over the dual K3 surface of degree two. We deduce that $\phi$ is an instance of a Mukai flop.