Invariant Metrics on Nilpotent Lie algebras

We state criteria for a nilpotent Lie algebra $\text{g}$ to admit an invariant metric. We use that $\text{g}$ possesses two canonical abelian ideals $\text{ide}(\text{g})\subset \mathfrak{J}(\text{g})$ to decompose the underlying vector space of $\text{g}$ and then we state sufficient conditions for $\text{g}$ to admit an invariant metric. The properties of the ideal $\mathfrak{J}(\text{g})$ allows to prove that if a current Lie algebra $\text{g}\otimes \text{Sa}$ admits an invariant metric, then there must be an invariant and non-degenerate bilinear map from $\text{Sa}\times \text{Sa}$ into the space of centroids of $\text{g}\mathrm{/}\mathfrak{J}(\text{g})$. We also prove that in any nilpotent Lie algebra $\text{g}$ there exists a non-zero, symmetric and invariant bilinear form. This bilinear form allows to reconstruct $\text{g}$ by means of an algebra with unit. We prove that this algebra is simple if and only if the bilinear form is an invariant metric on $\text{g}$.