The notion of an $\{i,j\}$-neutral operation of an $n$-grupoid has been introduced in [5], as a generalization of the neutral element in a grupoid (1.2). Every $n$-group [1, 1.3], $n\in|N\\backslash\{1\}$, has (uniquely determined 1.2.2) $\{1,n\}$-neutral operation [5, 1.3.2]. The condition $\{i,j\}\neq\{1,n\}$ is fulfilled for $n\geq3$; for $n=2$ the equality holds. In the present article, is, among others, given a necessary and sufficient condition for an $n$-group $(n\geq3)$ to have an $\{i,j\}$-neutral operation with the condition $\{i,j\}\neq\{1,n\}$ (Theorem 2.1).