  We introduce a series of $\Z_2^n$-graded quasialgebras $\bbP_n(m)$ which
generalizes Clifford algebras, higher octonions, and higher Cayley algebras.
The constructed series of algebras and their minor perturbations are applied to
contribute explicit solutions to the Hurwitz problem on compositions of
quadratic forms. In particular, we provide explicit expressions of the
well-known Hurwitz-Radon square identities in a uniform way, recover the
Yuzvinsky-Lam-Smith formulas, confirm the third family of admissible triples
proposed by Yuzvinsky in 1984, improve the two infinite families of solutions
obtained recently by Lenzhen, Morier-Genoud and Ovsienko, and construct several
new infinite families of solutions.
