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  "body": "\n## Definition\n\nLet $k$ be an algebraically closed field, and let $G$ be a [[linear algebraic group|GA81VVQHEP]] over $k$.\nA *Borel subgroup* of $G$ is a subgroup $B \\subseteq G$ such that:\n\n1. $B$ is closed in the [[Zariski topology|8SQK7RTB63]];\n2. $B$ is connected and [[solvable|0FXJ7YFMMX]];\n3. $B$ is maximal among the subgroups with properties 1 and 2.\n\n## Remarks\n\nOver an arbitrary field $k$, a closed subgroup $B$ of $G$ is called a Borel subgroup if $B \\times_k \\bar{k}$ is a Borel subgroup of $G \\times_k \\bar{k}$.\n\nExample: in $G = \\mathrm{GL}_n$ the group $B$ of invertible upper triangular matrices is a Borel subgroup.\nIt is closed, cut out by $x_{ij} = 0$ for $i \u003e j$.\nIt is connected and solvable, since its commutator subgroup consists of unipotent upper triangular matrices.\nIt is maximal: by the Lie–Kolchin theorem, a connected solvable subgroup $B' \\supseteq B$ fixes some full flag, and the only full flag fixed by $B$ is the standard one, so $B' \\subseteq B$.\n\nNon-example: in $\\mathrm{GL}_2$, the diagonal torus $T$ is closed, connected and solvable, but it fails clause 3, since $T \\subsetneq B$ for the upper triangular group $B$.\n\nAll Borel subgroups of $G$ are conjugate, and $G / B$ is a projective variety.\n",
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