  The fake monster Lie algebra is determined by the Borcherds function Phi_{12}
which is the reflective modular form of the minimal possible weight with
respect to O(II_{2,26}). We prove that the first non-zero Fourier-Jacobi
coefficient of Phi_{12} in any of 23 Niemeier cusps is equal to the Weyl-Kac
denominator function of the affine Lie algebra of the root system of the
corresponding Niemeier lattice. This is an automorphic answer (in the case of
the fake monster Lie algebra) on the old question of I. Frenkel and A. Feingold
(1983) about possible relations between hyperbolic Kac-Moody algebras, Siegel
modular forms and affine Lie algebras.
