Projective geometry from Poisson algebras

In analogy with the Poisson algebra of the quadratic forms on the symplectic plane, and the notion of duality in the projective plane introduced by Arnold in \cite{Arn}, where the concurrence of the triangle altitudes is deduced from the Jacobi identity, we consider the Poisson algebras of the first degree harmonics on the sphere, the pseudo-sphere and on the hyperboloid, to obtain analogous duality notions and similar results for the spherical, pseudo-spherical and hyperbolic geometry. Such algebras, including the algebra of quadratic forms, are isomorphic, as Lie algebras, either to the Lie algebra of the vectors in ${\mathbb{R}}^3$, with vector product, or to algebra $s{l}_2(\mathbb{R})$. The Tomihisa identity, introduced in \cite{Tom} for the algebra of quadratic forms, holds for all these Poisson algebras and has a geometrical interpretation. The relation between the different definitions of duality in projective geometry inherited by these structures is shown.