Subalgebras of the Split Octonions

Unlike the (ordinary) octonions, the split octonions have subalgebras that are not themselves composition algebras. The composition property itself doesn't fail; if it holds in the full algebra, it holds in all subalgebras. Rather, it is the requirement that the norm be nondegenerate that fails; these subalgebras are all null (to various degrees).

We have already seen that $\frac12(1\pm L)$ are projection operators in $\CC'$. This means that the subalgebra $\langle 1+L \rangle$ (consisting of all real multiples of $1+L$) closes under multiplication, and is therefore a subalgebra of $\CC'\subset\OO'$. This subalgebra is isomorphic to the real numbers, since $\frac12(1+L)$ acts as an identity element, but is not isometric to the real numbers, since $|1+L|=0$. All elements of this subalgebra are null, that is, have norm zero.

Other null elements of $\OO'$ also generate 1-dimensional null subalgebras, such as $\langle I+IL \rangle$. In this case, not only is $|I+IL|=0$, but also $(I+IL)^2=0$; all products in this subalgebra are zero.

We can combine such null subalgebras in various ways. The 2-dimensional subalgebra $\langle I+IL,J-JL \rangle$ again has all products zero, whereas the 2-dimensional subalgebra $\langle 1+L,I+IL \rangle$ does not. This latter algebra has the peculiar property that there are elements whose product is zero in one order, but not the other, since \begin{equation} (1+L)(I+IL) = 0 \qquad (I+IL)(1+L)=2(I+IL) \end{equation} Similarly, there are 3-dimensional null subalgebras $\langle I+IL,J+JL,K-KL \rangle$ and $\langle 1+L,I+IL,J-JL \rangle$, and a 4-dimensional null subalgebra $\langle 1+L,I+IL,J+JL,K-KL \rangle$.

Each of the subalgebras above is totally null; every element has norm zero. There are also subalgebras of $\OO'$ that are only partially null; each such subalgebra contains the identity element $1$.

The best-known of these subalgebras are the 3-dimensional ternions, generated by $\{1,L,I+IL\}$, and the 6-dimensional sextonions, generated by $\{1,I,IL,L,J+JL,K-KL\}$; there is also a 4-dimensional subalgebra generated by $\{1,L,I+IL,K-KL\}$.

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