We begin with a review of the basics of the Yang algebra of noncommutative phase spaces and Born Reciprocal Relativity. A solution is provided for the exact analytical mapping of the non-commuting $ x^mu, p^mu$ operator variables (associated to an $8D$ curved phase space) to the canonical $ Y^A, Pi^A$ operator variables of a flat $12D$ phase space. We explore the geometrical implications of this mapping which provides, in the $classical$ limit, with the embedding functions $ Y^A (x,p), Pi^A (x,p) $ of an $8D$ curved phase space into a flat $12D$ phase space background. The latter embedding functions determine the functional forms of the base spacetime metric $ g_{mu u} (x,p) $, the fiber metric of the vertical space $h^{ab}(x,p)$, and the nonlinear connection $N_{a mu} (x,p) $ associated with the $8D$ cotangent space of the $4D$ spacetime. A review of the mathematical tools behind curved phase spaces, Lagrange-Finsler, and Hamilton-Cartan geometries follows. This is necessary in order to answer the key question of whether or not the solutions found for $ g_{mu u} , h^{ab}, N_{a mu}$ as a result of the embedding, also solve the generalized gravitational vacuum field equations in the $8D$ cotangent space. We finalize with an Appendix with the key calculations involved in solving the exact analytical mapping of the $ x^mu, p^mu$ operator variables to the canonical $ Y^A, Pi^A$ operator ones.