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  "body": "\n## Example\n\nLet $n \\ge 2$ and let $\\mathrm{Cl}_n$ be the real [[Clifford algebra|A38B1QFK20]] of $\\R^n$ with the relations $v^2 = -\\abs{v}^2$ for $v \\in \\R^n$.\nThe *spin group* $\\mathrm{Spin}(n)$ is the set of products $u_1 u_2 \\cdots u_{2k}$ in $\\mathrm{Cl}_n$ of an even number of unit vectors $u_i \\in \\R^n$, with the Clifford product.\nIt is a group: the product of two such elements is again one.\nThe inverse of $u_1 \\cdots u_{2k}$ is $u_{2k} \\cdots u_1$, because $u_i u_i = -1$ and the number of factors is even.\n\nFor $x \\in \\mathrm{Spin}(n)$ and $v \\in \\R^n$, the element $x v x^{-1}$ lies again in $\\R^n$.\nCheck for one unit vector $u$: $u^{-1} = -u$, and $uv = -vu$ when $v \\perp u$.\nSo $-u v u^{-1}$ equals $v$ for $v \\perp u$ and $-u$ for $v = u$, and $v \\mapsto -u v u^{-1}$ is the reflection $r_u$ of $\\R^n$ in the hyperplane $u^\\perp$.\nFor $x = u_1 \\cdots u_{2k}$ the signs cancel, and $v \\mapsto x v x^{-1}$ is the product $r_{u_1} \\cdots r_{u_{2k}}$ of $2k$ reflections, which lies in [[SO(n)|0CGYNDCWRR]].\n\nThis gives a homomorphism $\\rho \\colon \\mathrm{Spin}(n) \\to SO(n)$, $\\rho(x)(v) = x v x^{-1}$.\nIt is surjective, because every element of $SO(n)$ is a product of an even number of reflections, and its kernel is $\\set{1, -1}$.\n\n## Remarks\n\n$\\mathrm{Spin}(n)$ is a compact connected Lie group, and $\\rho$ is a two-sheeted [[covering map|3XZRRE165N]].\nFor $n \\ge 3$ the group $\\mathrm{Spin}(n)$ is simply connected, so it is the universal cover of $SO(n)$.\nFor $n = 2$ the group $\\mathrm{Spin}(2)$ is a circle, and $\\rho$ is the double cover $z \\mapsto z^2$ of the circle $SO(2)$.\nIn low dimensions there are isomorphisms $\\mathrm{Spin}(3) \\cong$ [[SU(2)|E5P8AG6CK7]], $\\mathrm{Spin}(4) \\cong SU(2) \\times SU(2)$, $\\mathrm{Spin}(5) \\cong \\mathrm{Sp}(2)$ and $\\mathrm{Spin}(6) \\cong SU(4)$.\nThe same construction for a quadratic form of signature $(p, q)$ gives the groups $\\mathrm{Spin}(p, q)$.\n\n## Explanations\n\n### L1 human\n\nThe spin group is a twofold cover of the rotation group: each rotation corresponds to exactly two elements of $\\mathrm{Spin}(n)$, which differ by a sign.\nIt is built from products of unit vectors in the Clifford algebra, in which a vector squares to minus its length squared.\nEach unit vector stands for a reflection, and a product of an even number of reflections is a rotation.\nPhysicists need the spin group because electrons and other spin $1/2$ particles change sign under a full turn, so they transform under $\\mathrm{Spin}(3)$ rather than $SO(3)$.\n",
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