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Abstract 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 BI BI, the moduli space of bi-invariant metrics, is an orbifold. Moreover, we give an explicit description of this orbifold, and of EBI EBI, the space of bi-invariant metrics equivalent under isometries and scalar multiplies.
Juan Flores Torres (2024) studied this question.