  In this work we define, for the first time, the affine and projective plane
over the real Okubo algebra, showing a concrete geometrical interpretation of
its Spin group. Okubo algebra is a flexible, composition algebra which is also
a not unital division algebra. Even though Okubo algebra has been known for
more than 40 years, we believe that this is the first time the algebra was used
for affine and projective geometry. After showing that all axioms of affine
geometry are verified, we define a projective plane over Okubo algebra as
completion of the affine plane and directly through the use of Veronese
coordinates. We then present a bijection between the two constructions. Finally
we show a geometric interpretation of Spin(O) as the group of collineations
that preserve the axis of the plane.
