High Quality Content by WIKIPEDIA articles! In mathematics, a Lie algebra g is solvable if its derived series terminates in the zero subalgebra. That is, writing [mathfrak{g},mathfrak{g}] for the derived Lie algebra of g, generated by the [x,y] for x and y in g, the derived series mathfrak{g} geq [mathfrak{g},mathfrak{g}] geq [[mathfrak{g},mathfrak{g}],[mathfrak{g},mathfrak{g}]] geq [[[mathfrak{g},mathfrak{g}],[mathfrak{g},mathfrak{g}]],[[mathfrak{g},mathfrak{g}],[mathfrak{g},mathfrak{g}]]] geq ... becomes constant eventually at 0. Any nilpotent Lie algebra is solvable, a fortiori, but the converse is not true. The solvable Lie algebras and the semisimple Lie algebras form two large and generally complementary classes, as is shown by the Levi decomposition.

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