Joins of ideals of subideals of Lie algebras (Q1071100)

Suppose that a Lie algebra \(L\) is the join of two subideals \(H\) and \(K\). Let \(A\) be an ideal of \(H\), and \(B\) an ideal of \(K\). When is the join of \(A\) and \(B\) a subideal of \(L\)? Results of \textit{B. Hartley} [Proc. Camb. Philos. Soc. 63, 257--272 (1967; Zbl 0147.28201)] show that this is not always the case. The main result of the paper is that it holds if either: (a) the underlying field has characteristic zero, \(L/L^ 2\) is finitely generated, and \(H/A\), \(K/B\) are finite-dimensional (or satisfy \(\text{Max} -\triangleleft^ 2\) or \(\text{Min} -\triangleleft^ 2\) (ascending (descending) chain condition for 2-step subideals)); (b) the field is arbitrary and \(H/A\), \(K/B\) are perfect. Some of the results are analogues of group-theoretic ones due to \textit{H. F. Smith} [Glasg. Math. J. 25, 103--105 (1984; Zbl 0534.20016)]. A number of examples are given to show that various hypotheses cannot be relaxed, nor conclusions strengthened. The proof is based on a theorem of \textit{J. P. Williams} [Ph. D. thesis, Cambridge 1982] on universal enveloping algebras.