  We present a Veronese formulation of the octonionic and split-octonionic
projective and hyperbolic planes. This formulation of the incidence planes
highlights the relationship between the Veronese vectors and the rank-1
elements of the Albert algebras over octonions and split-octonions, yielding to
a clear formulation of the relationship with the real forms of the Lie groups
arising as collineation groups of these planes. The Veronesean representation
also provides a novel and minimal construction of the same octonionic and
split-octonionic planes, by exploiting two symmetric composition algebras: the
Okubo algebra and the paraoctonionic algebra. Besides the intrinsic
mathematical relevance of this construction of the real forms of the
Cayley-Moufang plane, we expect this approach to have implications in all
mathematical physics related with exceptional Lie Groups of type $G_{2},F_{4}$
and $E_{6}$.
