- §1. Cayley-Dickson
- §2. Sedenions
- §3. The Hurwitz Theorem
- §4. Split Complex Numbers
- §5. Split Quaternions
- §6. Split Octonions
- §7. Subalgebras
Split Octonions
If we repeat this process one more time, we obtain the split octonions, denoted $\OO'$, which satisfies \begin{equation} \OO' = \HH \oplus \HH L \end{equation} where we now use $I$, $J$, $K$ for the imaginary units in $\HH$. Thus, $\OO'$ consists of linear combinations of $\{1,I,J,K,KL,JL,IL,L\}$, and it again remains to work out the full multiplication table; the result is shown in Table 1. The split octonions are not associative, but they are alternative.
It is easily checked that the inner product now has signature $(4,4)$; our conventions are such that basis elements containing $L$ have (squared) norm $-1$, and all others have (squared) norm $+1$.
The split octonions can also be obtained as \begin{equation} \OO' = \HH' \oplus \HH'J \end{equation} so there are again only two 8-dimensional composition algebras over the reals, namely $\OO$ and $\OO'$. As before, the split octonions $\OO'$ contain both the split quaternions $\HH'$ and the ordinary quaternions $\HH$.
$I$ | $J$ | $K$ | $KL$ | $JL$ | $IL$ | $L$ | |
---|---|---|---|---|---|---|---|
$I$ | $-1$ | $K$ | $-J$ | $JL$ | $-KL$ | $-L$ | $IL$ |
$J$ | $-K$ | $-1$ | $I$ | $-IL$ | $-L$ | $KL$ | $JL$ |
$K$ | $J$ | $-I$ | $-1$ | $-L$ | $IL$ | $-JL$ | $KL$ |
$KL$ | $-JL$ | $IL$ | $L$ | $1$ | $-I$ | $J$ | $K$ |
$JL$ | $KL$ | $L$ | $-IL$ | $I$ | $1$ | $-K$ | $J$ |
$IL$ | $L$ | $-KL$ | $JL$ | $-J$ | $K$ | $1$ | $I$ |
$L$ | $-IL$ | $-JL$ | $-KL$ | $-K$ | $-J$ | $-I$ | $1$ |
Table 1: The split octonionic multiplication table.