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  "body": "\n## Definition\n\nLet $\\mathfrak{g}$ be a simple Lie algebra over $\\C$ with Cartan subalgebra $\\mathfrak{h}$ and root system $\\Phi$.\nLet $\\set{h_1, \\dots, h_\\ell} \\cup \\set{e_\\alpha : \\alpha \\in \\Phi}$ be a Chevalley basis of $\\mathfrak{g}$, and let $\\mathfrak{g}_\\Z$ be its $\\Z$-span.\nLet $K$ be a field and let $\\mathfrak{g}_K = \\mathfrak{g}_\\Z \\otimes_\\Z K$.\nFor $\\alpha \\in \\Phi$ and $t \\in K$, let $x_\\alpha(t) = \\sum_{k \\ge 0} t^k (\\operatorname{ad} e_\\alpha)^k / k!$, an automorphism of $\\mathfrak{g}_K$.\nHere $(\\operatorname{ad} e_\\alpha)^k / k!$ is the reduction of an integer matrix on $\\mathfrak{g}_\\Z$ and is $0$ for $k \\ge 4$.\nThe (adjoint) *Chevalley group* of type $\\Phi$ over $K$ is the subgroup of the [[automorphism group|0K1SCPBNKJ]] of $\\mathfrak{g}_K$ generated by all these $x_\\alpha(t)$.\n\n## Remarks\n\nIt is written $\\Phi(K)$, for example $A_n(K)$, $E_8(K)$ or $G_2(q)$ when $K = \\F_q$.\nThe sum defining $x_\\alpha(t)$ is finite because $\\operatorname{ad} e_\\alpha$ is nilpotent, and the Chevalley basis makes its divided powers integral, so the formula makes sense in every characteristic.\nExample: for $\\Phi$ of type $A_1$, the group $A_1(K)$ is isomorphic to $\\mathrm{PSL}_2(K)$.\nNon-example: let $K$ be a field of characteristic not $2$.\nThen $\\mathrm{SL}_2(K)$ is not the adjoint Chevalley group of type $A_1$.\nIts centre $\\set{\\pm I}$ has order $2$, while $A_1(K) \\cong \\mathrm{PSL}_2(K)$ has trivial centre.\nChevalley proved that $\\Phi(K)$ is a [[simple group|0YMYMNQ0QZ]] except for $A_1(\\F_2)$, $A_1(\\F_3)$, $B_2(\\F_2)$ and $G_2(\\F_2)$.\n\n## Explanations\n\n### L1 human\n\nA Chevalley group is the group generated by exponentials of root vectors of a simple Lie algebra, rebuilt over an arbitrary field.\n",
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