$Z_n$-Orthograded Monocomposition Algebras

We study NQM algebras $A$ having an orthogonal automorphism $\varphi$ of finite order $n\geqslant3 $ (called $Z_n$-orthograded NQM algebras). The $Z_3$-orthograded NQM algebras of dimension 7 are treated in more detail. In particular, we find all algebras $A$ which are not bi-isotropic in this class, and for every algebra $A$, determine an automorphism group $\operatorname{Aut}A$ and an orthogonal automorphism group $\operatorname{Ortaut}A$. In constructing and classifying (up to isomorphism) NQM algebras, use is made of orthogonal decompositions of the algebras.