Real octonions by Cayley–Dickson doubling

As a real vector space, set $\mathbb{O}:=\mathbb{H}\times \mathbb{H}$, with componentwise addition and real scalar multiplication, using the quaternion algebra, conjugation, and norm of The quaternions $\mathbb{H}$: real quadruples with componentwise addition and an explicit multiplication formula matching the table on $1,i,j,k$. Define

$(a,b)(c,d):=(ac-\bar{d}b,da+b\bar{c}),\overline{(a,b)}:=(\bar{a},-b)\mathrm{.}$

Distinguish $1_{\mathbb{O}}:=(1,0)$; the two-sided unit property is verified in The Cayley–Dickson octonions are alternative but not associative ↗. Write

$i=(i,0),j=(j,0),k=(k,0),\mathrm{\ell }=(0,1),i\mathrm{\ell }=(0,i),j\mathrm{\ell }=(0,j),k\mathrm{\ell }=(0,k)\mathrm{.}$

This fixes the multiplication convention.

The following directed cycles record the convention on the seven imaginary basis units. An oriented cycle $x\to y\to z\to x$ means $xy=z$, $yz=x$, and $zx=y$; reversing a product changes its sign. These cycles are an aid for reading the multiplication table, not a proof of alternativity or nonassociativity.

For example, the first cycle gives $ij=k$; the second gives $i\mathrm{\ell }=(0,i)$, $\mathrm{\ell }(i\mathrm{\ell })=i$, and, on reversing the first product, $\mathrm{\ell }i=-i\mathrm{\ell }$. These identities can also be checked directly from the pair formula. The proof of the alternative laws uses that formula, not this diagram. The theorem The Cayley–Dickson octonions are alternative but not associative ↗ proves both alternative laws and displays a nonassociative triple; neither property is assumed in this definition.