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 bypass 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