= Supersolvable Lie algebra
= Supersoluble Lie algebra
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= Supersolubility of a Lie algebra
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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.
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