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  "body": "\n## Description\n\nLet $\\mathbb{H}$ be the [[quaternions|0GE4DVQKR8]], with basis $1, i, j, k$ over $\\R$ and $i^2 = j^2 = k^2 = ijk = -1$.\nThe *Hurwitz quaternions* are the elements\n$$a + bi + cj + dk \\in \\mathbb{H}$$\nsuch that either $a, b, c, d$ are all in $\\Z$ or all in $\\Z + \\tfrac{1}{2}$.\nThey form a ring, written $\\mathcal{H}$ here.\nEquivalently, $\\mathcal{H}$ is the $\\Z$-span of $i$, $j$, $k$ and $\\omega = \\tfrac{1}{2}(1 + i + j + k)$.\n\n## Remarks\n\nThe ring $\\mathcal{H}$ is a maximal [[order|1ZBX4TKVZ9]] in the rational [[quaternion algebra|ZWXW0T3EAC]] $(-1, -1)_{\\Q}$.\nIt contains the Lipschitz quaternions, those with $a, b, c, d \\in \\Z$, as a subring of index $2$; the Lipschitz quaternions are an order but not a maximal one.\nThe reduced norm $a^2 + b^2 + c^2 + d^2$ takes integer values on $\\mathcal{H}$; for example $\\omega$ has norm $4 \\cdot \\tfrac14 = 1$.\nThe units of $\\mathcal{H}$ are the $24$ elements $\\pm 1, \\pm i, \\pm j, \\pm k$ and $\\tfrac12(\\pm 1 \\pm i \\pm j \\pm k)$; they form the binary tetrahedral group.\nHurwitz quaternions admit division with remainder on each side, so every one-sided ideal of $\\mathcal{H}$ is principal.\nThis is a standard route to [[Lagrange's four-square theorem|0KYYSMEVHH]].\n\n## Explanations\n\n### L1 human\n\nThe Hurwitz quaternions are the quaternion analogue of the integers.\nOne allows all four coordinates to be integers, or all four to be halves of odd integers.\nAdding the half-integer points makes the set a better ring than the plain integer quaternions: one can divide with a small remainder, as for whole numbers.\nThe quaternion $\\tfrac12(1 + i + j + k)$ is a typical new element, and it is even a unit, with norm $1$.\n",
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