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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a Lie algebra is reductive if its adjoint representation in completely reducible, whence the name. More concretely, a Lie algebra is reductive if is a direct sum of a semisimple Lie algebra and an abelian Lie algebra: mathfrak{g} = mathfrak{s} oplus mathfrak{a}; there are alternative characterizations, given below.The most basic example is the Lie algebra mathfrak{gl}_n of n times n matrices with the commutator as Lie bracket, or more abstractly as the endomorphism algebra of an n-dimensional vector space, mathfrak{gl}(V). This is the Lie algebra of the general linear group GL(n), and is reductive as it decomposes as mathfrak{gl}_n = mathfrak{sl}_n oplus mathfrak{k}, corresponding to traceless matrices and scalar matrices.