Lie algebras with complex structures having nilpotent eigenspaces (Q2891215)

Let \(\mathfrak g\) be a real Lie algebra with an almost complex structure \(J\) such that \(J[X,Y]-[JX,Y]-[X,JY]-J[JX,JY] = 0\). Let \({\mathfrak g}_*\) be \(\mathfrak g\) endowed with the algebra structure \((X,Y) \mapsto \frac{1}{2}([X,Y]-[JX,JY])\). The main result of this article states that if \({\mathfrak g}_*\) is nilpotent, then \(\mathfrak g\) is solvable. It works out all 6-dimensional nilpotent \({\mathfrak g}_*\). It also provides some related examples of complex and hypercomplex structures on solvable Lie algebras.