  In this work we present a useful way to introduce the octonionic projective
and hyperbolic plane through the use of Veronese vectors. Then we focus on
their relation with the exceptional Jordan algebra and show that the Veronese
vectors are the rank-one elements of the algebra. We then study groups of
motions over the octonionic plane recovering all real forms of G2, F4and E6
groups and finally give a classification of all octonionic and split-octonionic
planes as symmetric spaces.
