A unital in the generalized hexagon of order two, and an exceptional isomorphism between finite groups of Lie type

We construct a model of the Hermitian unital of order 3 (obtained from the non-degenerate hermitian form in three variables over the field of order 9) inside the octonion algebra over the field of order 2. This construction is invariant under the automorphism group of that algebra, and explains the known isomorphism from the finite group of exceptional Lie type onto the group of semi-similitudes of the hermitian form.