Given a general octonion x:
$\mathbb{O}$=$\mathbb{H}$+$\mathbb{H}$$L$ by
x=$x^1$+$x^2$i+$x^3$j+$x^4$k+$x^5$i$L$+$x^6$j$L$+$x^7$k$L$+$x^8$$L$
with $L^2$=-1,
and a general split-octonion x:
$\mathbb{O'}$=$\mathbb{H}$+$\mathbb{H}$$l$ by
x=$x^1$+$x^2$i+$x^3$j+$x^4$k+$x^5$i$l$+$x^6$j$l$+$x^7$k$l$+$x^8$$l$
with $l^2$=1,
does $L$ anti-commute or commute with $l$?
That is, do we have {$L$,$l$}=0 or do we have [$L$,$l$]=0?
How is the result proven?