Free nilpotent and H-type Lie algebras. Combinatorial and orthogonal designs

The aim of our paper is to construct pseudo H-type algebras from the covering free nilpotent two-step Lie algebra as the quotient algebra by an ideal. We propose an explicit algorithm of construction of such an ideal by making use of a non-degenerate scalar product. Moreover, as a byproduct result, we recover the existence of a rational structure on pseudo H-type algebras, which implies the existence of lattices on the corresponding pseudo H-type Lie groups. Our approach substantially uses combinatorics and reveals the interplay of pseudo H-type algebras with combinatorial and orthogonal designs. One of the key tools is the family of Hurwitz–Radon orthogonal matrices.