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  "body": "\n## Definition\n\nLet $G$ be a connected real or complex [[Lie group|7JBRGCVJRB]], and let $\\mathfrak{g}$ be its Lie algebra.\nThe group $G$ is a *semisimple Lie group* if $\\mathfrak{g}$ is a semisimple Lie algebra, that is, the only solvable ideal of $\\mathfrak{g}$ is $0$.\n\n## Remarks\n\nLet $\\mathfrak{g}$ be a finite-dimensional Lie algebra over a field of characteristic $0$.\nThe following are equivalent:\n\n1. the only solvable ideal of $\\mathfrak{g}$ is $0$;\n2. $\\mathfrak{g}$ is a direct sum of simple ideals;\n3. the Killing form $B(x, y) = \\tr(\\operatorname{ad} x \\operatorname{ad} y)$ is nondegenerate (Cartan's criterion).\n\nExample: $G = SO(4)$.\nIts Lie algebra $\\mathfrak{so}(4)$ is isomorphic to $\\mathfrak{so}(3) \\oplus \\mathfrak{so}(3)$, a direct sum of two simple ideals, so $G$ is semisimple.\nIt is not simple, since each summand is a proper nonzero ideal.\n\nNon-example: $G = GL_n(\\R)^+$, the invertible real $n \\times n$ matrices with positive determinant, for $n \\ge 1$.\nIts Lie algebra $\\mathfrak{gl}_n(\\R)$ contains the ideal $\\R I_n$ of scalar matrices, which is abelian, hence solvable, and nonzero.\nSo the condition fails.\n\nSome authors do not require $G$ to be connected and ask only that $\\mathfrak{g}$ be semisimple; some also ask for a finite centre.\n",
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