Let (g, [·,]) be a Lie algebra with an integrable complex structure J. The ±i eigenspaces of J are complex subalgebras of gC isomorphic to the algebra (g, [*]J) with bracket [X * Y]J = 1/2 ([X, Y] - [JX, J Y]). We consider here the case where these subalgebras are nilpotent and prove that the original (g, [·,]) Lie algebra must be solvable. We consider also the 6-dimensional case and determine explicitly the possible nilpotent Lie algebras (g, [*]J). 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 (g, [*]J) is s-step nilpotent. Similar examples of hypercomplex structures are also built.302247263Andrada, A., Barberis, M.L., Dotti, I.G., Ovando, G.P., Product Structures on Four Dimensional Solvable Lie algebras (2005) Homology, Homotopy and Applications, 7, pp. 9-37Barberis, M.L., Dotti, I., Abelian Complex Structures on Solvable Lie Algebras (2004) J. Lie Theory, 14, pp. 25-34Bartolomeis, P., (2005) Z2 and Z-Deformation Theory for Holomorphic and Symplectic Manifolds, pp. 75-103. , Complex, contact and symmetric manifolds, Progr. Math., 234 Birkhäuser Boston, Boston, MABorel, A., Groupes Linéaires Algébriques (1956) Ann. of Math, 64, pp. 20-82Cordero, L.A., Fernandez, M., Gray, A., Ugarte, L., Nilpotent Complex Structures (2001) Rev. R. Acad. Cien. Serie A. Mat, 95 (1), pp. 45-55Dotti, I., Fino, A., Abelian Hypercomplex 8-Dimensional Nilmanifolds (2000) Ann. Global Anal. Geom, 18, pp. 47-59Dotti, I., Fino, A., Hypercomplex Eight-Dimensional Nilpotent Lie Groups (2003) J. Pure Appl. Algebra, 184, pp. 41-57Dotti, I., Fino, A., Hypercomplex Nilpotent Lie Groups (2001) Contemp. Math, 288, pp. 310-314Dotti, I., Fino, A., HyperKahler Torsion Structures Invariant by Nilpotent Lie Groups (2002) Class. Quantum Grav., 19, pp. 551-562Goto, M., Note on a Characterization of Solvable Lie Algebras (1962) J. Sci. Hiroshima Univ. Ser. A-I, 26, pp. 1-2Michael, A.A.G., On the Conjugacy Theorem of Cartan Subalgebras (2002) Hiroshima Math. J., 32, pp. 155-163Petravchuk, P., Lie algebras Decomposable as Sum of an Abelian and a Nilpotent subalgebra (1988) Ukr. Math. J., 40, pp. 385-388Salamon, S., Complex Structures on Nilpotent Lie Algebras (2001) J. Pure appl. Algebra, 157, pp. 311-333San Martin, L.A.B., (1999) álgebras de Lie, , Editora da UnicampSerre, J.P., (1987) Complex Semisimple Lie Algebras, , Springer-Verlag, New York