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  "body": "\n## Statement\n\nLet $G$ be a [[compact Lie group|9TFC6ADC1C]], and let $\\mathfrak{g}$ be its Lie algebra, with center $\\mathfrak{z}$ and Killing form $B$.\nThen:\n\n1. $\\mathfrak{g}$ is a reductive Lie algebra: $\\mathfrak{g} = \\mathfrak{z} \\oplus [\\mathfrak{g}, \\mathfrak{g}]$ as Lie algebras, and $[\\mathfrak{g}, \\mathfrak{g}]$ is semisimple;\n2. $B(X, X) \\le 0$ for all $X \\in \\mathfrak{g}$, with equality exactly when $X \\in \\mathfrak{z}$.\n\n## Remarks\n\nThe proof averages an inner product on $\\mathfrak{g}$ over $G$ with the normalized [[Haar measure|VY0HXD3GC4]].\nThis gives an inner product invariant under $\\mathrm{Ad}(G)$, so each $\\operatorname{ad} X$ is skew-symmetric for it.\nThe orthogonal complement of an ideal is then an ideal, so $\\mathfrak{g}$ is a direct sum of simple ideals and of the abelian ideal $\\mathfrak{z}$.\nFor 2, the eigenvalues of the skew-symmetric map $\\operatorname{ad} X$ are purely imaginary, so $B(X, X) = \\tr\\big((\\operatorname{ad} X)^2\\big) \\le 0$, with equality only when $\\operatorname{ad} X = 0$.\n\nExample: for $G = \\mathrm{U}(n)$, the Lie algebra $\\mathfrak{u}(n)$ of skew-Hermitian matrices is $i\\R \\cdot I \\oplus \\mathfrak{su}(n)$, the center plus a simple ideal for $n \\ge 2$.\n\nCompactness is needed.\nThe group of affine maps $x \\mapsto ax + b$ of $\\R$ with $a \u003e 0$ is a noncompact Lie group whose Lie algebra is solvable but not abelian, so it is not reductive.\n\n## Explanations\n\n### L1 human\n\nThe infinitesimal version of a compact group is very well behaved.\nIt splits into a commutative part and a part made of simple pieces, with no mixing between them.\nThe reason is that a compact group can be averaged over, which produces a symmetric inner product that the group respects.\n\n### L2 human\n\nThe theorem is the first step of the structure theory of compact Lie groups.\nA real Lie algebra with an inner product for which each $\\operatorname{ad} X$ is skew-symmetric is called a compact Lie algebra.\nThe compact Lie algebras are exactly the reductive ones with negative semidefinite Killing form.\nConversely, each semisimple Lie algebra with negative definite Killing form is the Lie algebra of a compact Lie group, by Weyl's theorem that its simply connected group is compact.\nTogether with Weyl's unitary trick, this lets one move between compact groups and complex semisimple Lie algebras.\n",
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