Lie algebras in which every 1-dimensional subideal is an ideal (Q1098924)

A Lie algebra in which every subideal is an ideal is called a \(t\)-algebra. If any subalgebra of \(L\) is a \(t\)-algebra then \(L\) is called a \(T\)-algebra and if every nilpotent subideal of \(L\) is an ideal of \(L\) then \(L\) is called a \(c\)-algebra. The authors first prove that if \(L\) is a serially finite Lie algebra over a field of characteristic zero and if the locally soluble radical of \(L\) belongs to the class É(si)\({\mathfrak A}\) of Lie algebras then the following are equivalent: (i) \(L\) is a \(c\)-algebra. (ii) \(L\) is a \(t\)-algebra. (iii) \(L=R\oplus S\) where \(R\) is an ideal of \(L\) which is either abelian or almost abelian and S isa semisimple ideal of \(L\). It is also shown that for any Lie algebra the following are equivalent: (i) \(L\) is a \(c\)-algebra. (ii) Every subalgebra of \(L\) is a \(c\)-algebra. (iii) Every 1-dimensional ascending subalgebra of a subalgebra \(H\) of \(L\) is an ideal of \(H\). It is proved that for a locally finite Lie algebra \(L\) over any field the following are equivalent: (i) \(L\) is a \(C\)-algebra. (ii) \(L\) is a \(T\)-algebra. (iii) Every serial subalgebra \(H\) of \(L\) is an ideal of \(H\). (iv) Every 1- dimensional serial subalgebra of a subalgebra \(H\) of \(L\) is an ideal of \(H\). Finally the authors show by giving several examples that over any field there exists a \(c\)-algebra which is neither a \(C\)-algebra nor a \(t\)-algebra and a \(t\)-algebra which is not a \(T\)-algebra. They also show that over any field of \(\text{char}=0\) there exists a \(C\)-algebra which is not a \(T\)-algebra.