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  "body": "\n## Description\n\nLet $n \\ge 1$.\nThe *special orthogonal group* is the set\n$$SO(n) = \\{Q \\in O(n) : \\det Q = 1\\}$$\nof real orthogonal $n \\times n$ matrices with [[determinant|00KDZRW4C0]] $1$, with matrix multiplication.\nIt is a normal subgroup of index $2$ of the [[orthogonal group|0TKYAEJ8N7]] $O(n)$.\n\n## Remarks\n\nIts elements are the *rotations* of $\\R^n$ about the origin.\n$SO(2)$ consists of the [[rotation matrices|0B9QNFGTHM]] $\\begin{pmatrix} \\cos\\theta \u0026 -\\sin\\theta \\\\ \\sin\\theta \u0026 \\cos\\theta \\end{pmatrix}$, and it is commutative.\nEach element of $SO(3)$ is a rotation about an axis through the origin.\n$SO(n)$ is a connected compact Lie group.\n\n## Explanations\n\n### L1 human\n\nThe special orthogonal group contains the orthogonal matrices that do not flip orientation.\nThese are the rotations.\nReflections have determinant $-1$, so they are not in this group.\n",
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