An n-quasigroup $(Q,f)$ is reducible if it can be represented by two quasigroups of smaller arities, and it is completely reducible if it can be represented by $n-1$ binary quasigroups. It is proved that if an $A_n$-commutative $n$-quasigroup is completely reducible, where $A_n$ is the alternating subgroup of the symmetrtic group $S_n$ of degree $n$, then there exists an Abelian group to which all component binary quasi-groups are isotopic. It is also proved that for every subgroup $H\subseteq S_n$ there exists a nonreducible exactly $H$-commutative $n$-quasigroup of every composite order $mp$, where $m>2$, $p\geq n\geq 3$.