Solvable Lie algebra

Summary

A solvable Lie algebra is a Lie algebra whose derived series reaches $0$. The derived series is the chain of iterated brackets $\mathfrak g \supseteq [\mathfrak g, \mathfrak g] \supseteq [\,[\mathfrak g, \mathfrak g], [\mathfrak g, \mathfrak g]\,] \supseteq \cdots$. It is the Lie algebra analogue of a solvable group. Algebras of upper triangular matrices are the model examples. By Lie's theorem, over an algebraically closed field of characteristic $0$, every solvable algebra of matrices is of this kind. Over a field of characteristic $0$, every finite-dimensional Lie algebra is built from a solvable part, its radical, and a semisimple part.