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- The okubic projective plane in its relation with the cayley planePublication . Daniele, Corradetti; Manojlovic, Nenad; Zucconi, FrancescoIn this dissertation, we delve into algebraic and geometrical applications of 8-dimensional composition algebras, examining their relationships with exceptional Jordan algebras, exceptional Lie groups, and classical geometric constructs such as the Cayley projective plane. Our main focus is on the Okubo algebra, which is a non-associative composition and division algebras that lacks a unit element. After a brief review on composition algebras, we introduce octonionic projective planes using a generalised version of Veronese conditions in order to show a straightforward way to relate them to Albert algebras and real forms of exceptional Lie groups of type F4 and E6. We then use a modified version of such conditions that allows to define an affine and projective plane over the Okubo algebra. We also find that a similar construction also holds for the para-octonionic algebra which is again an 8-dimensional composition algebra that lacks of unit element, but is para-unital. In the ensuing sections, we present isomorphisms linking the Okubo projective plane with both the para-octonionic and octonionic projective planes. The result is surprising, given that para-octonions and the Okubo algebra neither exhibit alternativity nor conform to the Moufang identities. Historically, the compact 16-dimensional Moufang plane, also known as the Cayley plane, arose out of octonionic geometry. However, in this work we show that this plane can be defined in an equally clean, straightforward and more minimal way by means of two different division composition algebras. Notably, these algebras possess a reduced algebraic structure compared to the octonions and do not uphold the Moufang identities, which are traditionally linked to the Moufang attributes of the plane.