Lie Algebras with Complex Structures Having Nilpotent Eigenspaces

Let $ g $ be a Lie algebra with an integrable complex structure $J$. The eigenspaces of $J$ are complex subalgebras of ${g}^{C}$ isomorphic to the algebra $({g},[*]_{J}) $. We consider here the case where these subalgebras are nilpotent and prove that the original Lie algebra $(g,[,])$ must be solvable. We consider also the $6$-dimensional case and determine explicitly the possible nilpotent * Lie algebras. Finally we produce several examples illustrating different situations, in particular we show that for each given $s$ there exists $g$ with complex structure $J$ such that its *-algebra is $s$-step nilpotent. Similar examples of hipercomplex structures are also built.