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  "body": "\n## Description\n\nThe *octonions* $\\mathbb{O}$ form an $8$-dimensional real algebra.\nOne construction is the Cayley–Dickson construction applied to the [[quaternions|0GE4DVQKR8]] $\\mathbb{H}$.\nAn octonion is a pair $(a, b)$ of quaternions, with componentwise addition and the product\n$$(a, b)(c, d) = (ac - \\bar{d} b,\\; da + b \\bar{c}),$$\nwhere $\\bar{q}$ is the quaternion conjugate.\nThe conjugate of $(a, b)$ is $(\\bar{a}, -b)$, and the norm is $N(a, b) = \\abs{a}^2 + \\abs{b}^2$.\nThe real numbers sit in $\\mathbb{O}$ as the pairs $(r, 0)$ with $r \\in \\R$, and $(1, 0)$ is the unit.\n\n## Remarks\n\n$\\mathbb{O}$ is not associative, so it is a [[nonassociative ring|YP5TQ6WFVN]] and not a ring in the usual sense.\nLet $i, j, k$ be the quaternion units, identified with $(i, 0)$, $(j, 0)$, $(k, 0)$, and let $\\ell = (0, 1)$.\nThe product formula gives $(a, 0)\\,\\ell = (0, a)$ and $(a, 0)(0, d) = (0, da)$.\nHence $(ij)\\ell = k\\ell = (0, k)$, while $i(j\\ell) = (i, 0)(0, j) = (0, ji) = (0, -k)$, so $(ij)\\ell \\ne i(j\\ell)$.\n$\\mathbb{O}$ is alternative: $x(xy) = (xx)y$ and $(yx)x = y(xx)$ for all $x, y$, and any two elements generate an associative subalgebra.\nThe norm is multiplicative, $N(xy) = N(x) N(y)$, so $\\mathbb{O}$ is a normed division algebra.\nBy Hurwitz's theorem, $\\R$, $\\C$, $\\mathbb{H}$ and $\\mathbb{O}$ are the only normed division algebras over $\\R$ with a unit.\nThe automorphism group of $\\mathbb{O}$ is the compact exceptional Lie group $G_2$.\n\n## Explanations\n\n### L1 human\n\nThe real numbers, the complex numbers and the quaternions can each be doubled to get the next one: complex numbers are pairs of reals, quaternions are pairs of complex numbers.\nDoubling the quaternions once more gives the octonions, eight real coordinates.\nAt each step something is lost: the complex numbers lose the ordering, the quaternions lose commutativity, and the octonions lose associativity.\nYet one can still divide by any nonzero octonion, and the length of a product is the product of the lengths.\nAfter the octonions this stops: the next doubling has zero divisors.\n",
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