We study the structure of a metric $n$-Lie algebra $\mathcal { G}$ over the complex field $\mathbb C$. Let $\mathcal { G}= \mathcal S\oplus { \mathcal R}$ be the Levi decomposition, where $\mathcal R$ is the radical of $\mathcal { G}$ and $\mathcal S$ is a strong semisimple subalgebra of $\mathcal { G}$. Denote by $m(\mathcal { G})$ the number of all minimal ideals of an indecomposable metric $n$-Lie algebra and $\mathcal R^\bot$ the orthogonal complement of $R$. We obtain the following results. As $\mathcal S$-modules, $\mathcal R^{ \bot}$ is isomorphic to the dual module of $\mathcal { G} / \mathcal R.$ The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on $\mathcal { G}$ equals that of the vector space of certain linear transformations on $\mathcal { G}$; this dimension is greater than or equal to $m(\mathcal { G}) + 1$. The centralizer of $\mathcal R$ in $\mathcal G$ equals the sum of all minimal ideals; it is the direct sum of $\mathcal R^\bot$ and the center of $\mathcal { G}$. The sufficient and necessary condition for $\mathcal { G}$ having no strong semisimple ideals is that $\mathcal R^\bot \subseteq \mathcal R$.
Categories: math.RA