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This paper studies how many orthogonal bi-invariant complex structures existon a metric Lie algebra over the real numbers. Recently, it was shown thatirreducible Lie algebras which are additionally $2$-step nilpotent admit atmost one orthogonal bi-invariant complex structure up to sign. The main resultgeneralizes this statement to metric Lie algebras with any number ofirreducible factors and which are not necessarily $2$-step nilpotent. It statesthat there are either $0$ or $2^k$ such complex structures, with $k$ the numberof irreducible factors of the metric Lie algebra. The motivation for thisproblem comes from differential geometry, for instance to constructnon-parallel Killing-Yano $2$-forms on nilmanifolds or to describe the compactChern-flat quasi-K\"ahler manifolds.  The main tool we develop is the unique orthogonal decomposition intoirreducible factors for metric Lie algebras with no non-trivial abelian factor.This is a generalization of a recent result which only deals with nilpotent Liealgebras over the real numbers. Not only do we apply this fact to describe theorthogonal bi-invariant complex structures on a given metric Lie algebra, butit also gives us a method to study different inner products on a given Liealgebra, computing the number of irreducible factors and orthogonalbi-invariant complex structures for varying inner products.

Orthogonal bi-invariant complex structures on metric Lie algebras