A note on complement ideals of Lie algebras (Q1336498)

Let \(L\) be a not necessarily finite-dimensional Lie algebra over any field. In this note we shall give an affirmative answer to Question 1.7 of \textit{F. A. M. Aldosray} and \textit{I. Stewart} [ibid. 22, 1-13 (1992; Zbl 0766.17020)]: If \(L\) is semisimple and \(I\), \(J\) are complement ideals of \(L\), then \(I \cap J\) always a complement ideal of \(I\)? \(L\) is called semisimple if \(L\) has no non-abelian ideals. Recall that an ideal \(J\) of \(L\) is complement if there exists an ideal \(N\) of \(L\) such that \(J \cap N = 0\) and \(K \cap N \neq 0\) for any ideal \(K\) of \(L\) with \(J \nsubseteq K\).