 Supersolvable Lie algebra

A finite-dimensional Lie algebra is supersolvable if it has a full flag of ideals of a Lie algebra, increasing by one dimension at each step. It is solvable over every field: induct on dimension using the first one-dimensional ideal, which is abelian, and the quotient flag. Conversely over an algebraically closed field of characteristic zero, Lie's theorem applied to the adjoint representation supplies a one-dimensional ideal in every nonzero solvable Lie algebra. Induct on the quotient and lift its flag. A full ideal flag in $L/I$ lifts across a one-dimensional ideal $I$ simply by taking inverse images and prepending zero. Thus solvability and supersolvability agree over the complex numbers, although they need not over an arbitrary field.