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 $\HH$.

Table 1: The split octonionic multiplication table.