Siegel automorphic form corrections of some Lorentzian Kac-Moody Lie algebras

We find automorphic form corrections which are generalized Lorentzian Kac-Moody superalgebras without odd real simple roots for two elliptic Lorentzian Kac-Moody algebras of rank 3 with a lattice Weyl vector, and calculate multiplicities of their simple and arbitrary roots. These Kac-Moody algebras are defined by hyperbolic symmetrized generalized Cartan matrices [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] of rank 3. Both these algebras have elliptic type (i.e., their Weyl groups have fundamental polyhedra of finite volume in corresponding hyperbolic spaces) and have a lattice Weyl vector. The correcting automorphic forms are Siegel modular forms. The form corresponding to G₁ is the classical Siegel cusp form of weight 5 which is the product of ten even theta-constants. In particular we find an infinite product formula for this modular form.