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Using the arithmetical reduction suggested in the previous talk, I will prove the two new theorems stated there, viz., “the number of tritangents to a smooth plane sextic is at most 72”, and “the number of 2-planes in a smooth cubic 4-fold is at most 405” (joint with I. Itenberg and J.Ch. Ottem). To this end, we will embed the appropriately modified lattice of algebraic cycles to a Niemeier lattice and study certain configurations of square 4 vectors in the latter. I will try to explain the advantages of this approach and outline the principal techniques used in counting square 4 vectors.