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  "body": "\n## Definition\n\nLet $G$ be a real or complex [[Lie group|7JBRGCVJRB]], and let $\\mathfrak{g}$ be its Lie algebra, with centre $\\mathfrak{z}(\\mathfrak{g})$ and derived algebra $[\\mathfrak{g}, \\mathfrak{g}]$.\nThe group $G$ is a *reductive Lie group* if $\\mathfrak{g}$ is a reductive Lie algebra, that is:\n$$\\mathfrak{g} = \\mathfrak{z}(\\mathfrak{g}) \\oplus [\\mathfrak{g}, \\mathfrak{g}] \\quad \\text{and} \\quad [\\mathfrak{g}, \\mathfrak{g}] \\text{ is semisimple}.$$\n\n## Remarks\n\nOver a field of characteristic $0$, $\\mathfrak{g}$ is reductive exactly when its adjoint representation is completely reducible.\n\nExample: $G = GL_n(\\R)$ with $n \\ge 1$.\nHere $\\mathfrak{g} = \\mathfrak{gl}_n(\\R)$, whose centre is the scalar matrices $\\R I_n$ and whose derived algebra is $\\mathfrak{sl}_n(\\R)$, the trace-zero matrices.\nEvery matrix $X$ is uniquely $\\frac{\\tr X}{n} I_n + \\left(X - \\frac{\\tr X}{n} I_n\\right)$, so $\\mathfrak{g} = \\R I_n \\oplus \\mathfrak{sl}_n(\\R)$, and $\\mathfrak{sl}_n(\\R)$ is $0$ or simple, hence semisimple.\n\nNon-example: the group $B$ of invertible real upper triangular $2 \\times 2$ matrices.\nIts Lie algebra $\\mathfrak{b}$ of upper triangular matrices has centre $\\R I_2$ and derived algebra $\\mathfrak{n}$ of strictly upper triangular matrices.\nThen $\\mathfrak{z}(\\mathfrak{b}) \\oplus \\mathfrak{n}$ has dimension $2$, not $3$, and $\\mathfrak{n}$ is abelian and nonzero, so not semisimple; both conditions fail.\n\nMany authors add further conditions, for example that $G$ has finitely many connected components, or that $G$ is the group of real points of a reductive algebraic group.\nUnder the definition above, connected semisimple Lie groups and compact Lie groups are reductive.\n",
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