  Following Sen[arXiv:0911.1563], we study the counting of (`twisted') BPS
states that contribute to twisted helicity trace indices in four-dimensional
CHL models with N=4 supersymmetry. The generating functions of half-BPS states,
twisted as well as untwisted, are given in terms of multiplicative eta products
with the Mathieu group, M_{24}, playing an important role. These multiplicative
eta products enable us to construct Siegel modular forms that count twisted
quarter-BPS states. The square-roots of these Siegel modular forms turn out be
precisely a special class of Siegel modular forms, the dd-modular forms, that
have been classified by Clery and Gritsenko[arXiv:0812.3962]. We show that each
one of these dd-modular forms arise as the Weyl-Kac-Borcherds denominator
formula of a rank-three Borcherds-Kac-Moody Lie superalgebra. The walls of the
Weyl chamber are in one-to-one correspondence with the walls of marginal
stability in the corresponding CHL model for twisted dyons as well as untwisted
ones. This leads to a periodic table of BKM Lie superalgebras with properties
that are consistent with physical expectations.
