Octonions

Remarks

$\mathbb{O}$ is not associative, so it is a nonassociative ring and not a ring in the usual sense. Let $i, j, k$ be the quaternion units, identified with $(i, 0)$, $(j, 0)$, $(k, 0)$, and let $\ell = (0, 1)$. The product formula gives $(a, 0)\,\ell = (0, a)$ and $(a, 0)(0, d) = (0, da)$. Hence $(ij)\ell = k\ell = (0, k)$, while $i(j\ell) = (i, 0)(0, j) = (0, ji) = (0, -k)$, so $(ij)\ell \ne i(j\ell)$. $\mathbb{O}$ is alternative: $x(xy) = (xx)y$ and $(yx)x = y(xx)$ for all $x, y$, and any two elements generate an associative subalgebra. The norm is multiplicative, $N(xy) = N(x) N(y)$, so $\mathbb{O}$ is a normed division algebra. By Hurwitz's theorem, $\R$, $\C$, $\mathbb{H}$ and $\mathbb{O}$ are the only normed division algebras over $\R$ with a unit. The automorphism group of $\mathbb{O}$ is the compact exceptional Lie group $G_2$.