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, that is, metrics invariant under both left and right translations. We show that $\mathfrak{B}\mathfrak{I}$, the moduli space of bi-invariant metrics, is an orbifold. Moreover we give an explicit description of this orbifold, and of $\mathfrak{E}\mathfrak{B}\mathfrak{I}$, the space of bi-invariant metrics equivalent under isometries and scalar multiplies.