### 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.

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