  We construct a tower of arithmetic generators of the bigraded polynomial ring
J_{*,*}^{w, O}(D_n) of weak Jacobi modular forms invariant with respect to the
full orthogonal group O(D_n) of the root lattice D_n for 2\le n\le 8. This
tower corresponds to the tower of strongly reflective modular forms on the
orthogonal groups of signature (2,n) which determine the Lorentzian Kac-Moody
algebras related to the BCOV (Bershadsky-Cecotti-Ooguri-Vafa)-analytic
torsions. We prove that the main three generators of index one of the graded
ring satisfy a special system of modular differential equations. We found also
a general modular differential equation of the generator of weight 0 and index
1 which generates the automorphic discriminant of the moduli space of Enriques
surfaces.
