On subideals of the join of permutable Lie algebras (Q788815)

In this note we shall continue our investigation for subideals of the join of permutable Lie algebras [for the definition of these and other terms, see \textit{R. K. Amayo} and \textit{I. Stewart}, Infinite-dimensional Lie algebras (1974; Zbl 0302.17006)] generalizing our previous results (see the preceding review) to certain classes of infinite-dimensional Lie algebras. The main results are as follows: Let L be a Lie algebra over a field of characteristic zero and let A, H, K be subalgebras of L such that \(L=H+K\) and \(A\subseteq H\), \(A\subseteq K\). Then (a) If L is soluble-by-finite and A is a subideal of H, A is a subideal of K, then A is a subideal of L. (b) If L is ideally finite and A is ascendant in H, A is ascendant in K, then A is ascendant in L. We also obtain some analogues of theorems of \textit{H. Wielandt} [Arch. Math. 35, 1-7 (1980; Zbl 0413.20020)]. Let L be a Lie algebra over any field and let \(H_ 1\), \(H_ 2\), \(H_ 3\) be subalgebras of L such that \(L=<H_ 1,H_ 2,H_ 3>\). If \([H_ 1,H_ 2]\subseteq H_ 1\) and if \(H_ i\) is a subideal of \(<H_ i,H_ j>=H_ i+H_ j\) for all \(i,j=1,2,3\), then \(H_ 1\), \(H_ 2\) and \(H_ 3\) are subideals of L. Finally if L is an ideally finite Lie algebra over a field of characteristic zero and if \(H_ i\) is ascendant in \(<H_ i,H_ j>\) for all \(i,j=1,2,3\) and \(<H_ 1,H_ 2>\) is permutable with \(H_ 3\) then \(H_ 1\), \(H_ 2\) and \(H_ 3\) are ascendant in \(L=<H_ 1,H_ 2,H_ 3>\).