Self-duality in Dimensions 2n>4: Equivalence of Various Definitions, and an Upper Bound for p₂

We show that the self-duality defined in [Trautman, Int.J.Theor.Phys.,{\bf 16},561 (1977)] is equivalent to strong self-duality defined in [Bilge, Dereli and Kocak, Lett.Math.Phys., {\bf 36}, 301-309, (1996)] and we obtain an upper bound on $\int (p_2)^{n}$, where $p_2$ is the second Pontrjagin class of an $SO(N)$ bundle over a $8n$ dimensional manifold.