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

BKM Lie superalgebras from counting twisted CHL dyons