Moduli space of bi-invariant metrics

In this work, we focus on describing the space of bi-invariant metrics in a Lie group up to isometry. i.e, metrics invariant under both left and right translations. We show that \(\mathfrak {BI}\) $\mathfrak{BI}$ , the moduli space of bi-invariant metrics, is an orbifold. Moreover, we give an explicit description of this orbifold, and of \(\mathfrak {EBI}\) $\mathfrak{EBI}$ , the space of bi-invariant metrics equivalent under isometries and scalar multiplies.