Complements of intervals and prefrattini subalgebras of solvable Lie algebras (Q2838947)

Let \(L\) be a finite-dimensional solvable Lie algebra over a field and \(U\) a subalgebra of \(L\). Denote by \([U:L]\) the set of all subalgebras of \(L\) containing \(U\). One says that \([U:L]\) is complemented if, for every \(S\in [U:L]\), there exists \(T\in [U:L]\) such that \(S\cap T=U\) and the subalgebra generated by \(S\) and \(T\) coincides with \(L\). In this interesting paper the author studies the set \(\Omega(U,L)\) of all \(S\in [U:L]\) such that \([S:L]\) is complemented. In particular, he shows that for a class of metanilpotent Lie algebras, the minimal elements of \(\Omega(U,L)\) are conjugate in \(L\). These ideas are then used to introduce a Lie-theoretic analogue of a generalization of the prefrattini subgroups.