The compact 16-dimensional Moufang plane, i.e., the Cayley plane, has traditionally been defined through octonionic geometry. Indeed, the octonions, being the largest division Hurwitz algebra, provide the usual algebraic foundation for \(\mathbb {O}P^{2}\) , the Cayley projective plane. This plane is central to the geometric realizations of all exceptional Lie groups, forming links between non-associative algebra, projective geometry, and exceptional Lie structures. Surprisingly, it turns out that the Cayley plane can also be constructed from weaker algebraic frameworks than the octonions. By replacing octonions with para-octonions \(p\mathbb {O}\) or with the non-unital and non-alternative Okubo algebra \(\mathcal {O}\) , one obtains the same incidence structure and projective geometry. This result may come as a surprise, given the common belief that Moufang planes can be realized only through alternative algebras in which the Moufang identity holds. In fact, in the case of non-unital algebras, even symmetric composition algebras, such as para-octonions or Okubo algebras, can yield equivalent results.