Reductive Lie algebra

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In mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, hence the name. More concretely, a Lie algebra is reductive if it is a direct sum of a semisimple Lie algebra and an abelian Lie algebra: ${\displaystyle {\mathfrak {g}}={\mathfrak {s}}\oplus {\mathfrak {a}};}$ there are alternative characterizations, given below.