  In 1983, Feingold and Frenkel posed a question about possible relations
between affine Lie algebras, hyperbolic Kac-Moody algebras and Siegel modular
forms. In this paper we give an automorphic answer to this question and its
generalization. We classify hyperbolic Borcherds-Kac-Moody superalgebras whose
super-denominators define reflective automorphic products of singular weight on
lattices of type $2U \oplus L$. As a consequence, we prove that there are
exactly $81$ affine Lie algebras $g$ which have nice extensions to hyperbolic
BKM superalgebras for which the leading Fourier-Jacobi coefficients of
super-denominators coincide with the denominators of $g$. We find that 69 of
them appear in Schellekens' list of semi-simple $V_1$ structures of holomorphic
CFT of central charge $24$, while $8$ of them correspond to the $N=1$
structures of holomorphic SCFT of central charge $12$ composed of $24$ chiral
fermions. The last $4$ cases are related to exceptional modular invariants from
nontrivial automorphisms of fusion algebras. This clarifies the relationship of
affine Lie algebras, vertex algebras and hyperbolic BKM superalgebras at the
level of modular forms.
