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Reference.org
Reductive Lie algebra
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> is reductive if its <a href="/facts/Adjoint_endomorphism/bWG4c2su">adjoint representation</a> is <a href="/facts/Lie_algebra_representation/aaxnMxCR">completely reducible</a>, hence the name. More concretely, a Lie algebra is reductive if it is a <a href="/facts/Direct_sum_of_Lie_algebras/Y6PBJnjg">direct sum</a> of a <a href="/facts/Semisimple_Lie_algebra/m8jA3Bcf">semisimple Lie algebra</a> and an <a href="/facts/Abelian_Lie_algebra/n2gAUkNq">abelian Lie algebra</a>: g = s ⊕ a ; {\displaystyle {\mathfrak {g}}={\mathfrak {s}}\oplus {\mathfrak {a}};} there are alternative characterizations, given below. </p>