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  "body": "\n## Summary\n\nA reductive group is a connected [[linear algebraic group|GA81VVQHEP]] with no nontrivial connected normal unipotent subgroup.\nExamples are $\\mathrm{GL}_n$, $\\mathrm{SL}_n$, the special orthogonal groups $\\mathrm{SO}_n$, the symplectic groups, and the [[algebraic tori|66FQFAJNS4]].\nOver an algebraically closed field, reductive groups are classified by root data.\nIn characteristic $0$, they are the connected linear groups whose representations are all semisimple.\n\n## Explanations\n\n### L1 human\n\nInside the group of invertible upper triangular matrices sits the subgroup of those with all diagonal entries $1$.\nThat subgroup is unipotent and normal, and it makes the group hard to analyze.\nA reductive group is one with no such part: nothing unipotent hides in it as a normal subgroup.\nThis is the class of groups with a clean classification, much like compact Lie groups.\n",
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