The derived series of a Lie algebra is a sequence of subalgebras formed by iteratively taking the derived algebra, which is the commutator of the algebra with itself. This series is crucial for understanding the structure of the Lie algebra, particularly in distinguishing between solvable and non-solvable algebras. Each successive quotient of this series helps to analyze the properties of the algebra, providing insight into its solvability and potential nilpotency.
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