ℍ𝕪𝕡𝕖𝕣𝕔𝕠𝕞𝕡𝕝𝕖𝕩 𝕤𝕥𝕦𝕗𝕗/𝕊𝕖𝕩𝕥𝕠𝕟𝕚𝕠𝕟𝕤

Sextonionic Multiplication table

$\cdot$	$1$	${\acute {i}}$	${\acute {\varepsilon }}$	${\acute {\varepsilon }}{\acute {i}}$	${\acute {j}}$	$i_{1}$
$1$	$1$	${\acute {i}}$	${\acute {\varepsilon }}$	${\acute {\varepsilon }}{\acute {i}}$	${\acute {j}}$	$i_{1}$
${\acute {i}}$	${\acute {i}}$	$-1$	$-{\acute {\varepsilon }}{\acute {i}}$	${\acute {\varepsilon }}$	$-i_{1}$	${\acute {j}}$
${\acute {\varepsilon }}$	${\acute {\varepsilon }}$	${\acute {\varepsilon }}{\acute {i}}$	$0$	$0$	$-{\acute {\varepsilon }}$	${\acute {\varepsilon }}{\acute {i}}$
${\acute {\varepsilon }}{\acute {i}}$	${\acute {\varepsilon }}{\acute {i}}$	$-{\acute {\varepsilon }}$	$0$	$0$	${\acute {\varepsilon }}{\acute {i}}$	${\acute {\varepsilon }}$
${\acute {j}}$	${\acute {j}}$	$i_{1}$	${\acute {\varepsilon }}$	$-{\acute {\varepsilon }}{\acute {i}}$	$1$	${\acute {i}}$
$i_{1}$	$i_{1}$	$-{\acute {j}}$	$-{\acute {\varepsilon }}{\acute {i}}$	$-{\acute {\varepsilon }}$	$-{\acute {i}}$	$1$

Split-octonionic basis unit representation of nilpotent basis sextonions

• ${\acute {\varepsilon }}={\frac {\sqrt {2}}{2}}(j+j_{1})$
• ${\acute {\varepsilon }}{\acute {i}}={\frac {\sqrt {2}}{2}}(k_{1}-k)$

Mnemonic

Ordered triplets

A chain $a\rightarrow b\rightarrow c$ on the mnemonic implies $a\cdot b=c$ and $b\cdot a=-c$

• ${\acute {j}}\rightarrow {\acute {\varepsilon }}\rightarrow {\acute {\varepsilon }}$
• ${\acute {\varepsilon }}\rightarrow {\acute {i}}\rightarrow {\acute {\varepsilon }}{\acute {i}}$
• ${\acute {i}}\rightarrow i_{1}\rightarrow {\acute {j}}$
• ${\acute {\varepsilon }}{\acute {i}}\rightarrow i_{1}\rightarrow {\acute {\varepsilon }}$
• ${\acute {\varepsilon }}{\acute {i}}\rightarrow {\acute {j}}\rightarrow {\acute {\varepsilon }}{\acute {i}}$

Graph data

https://csacademy.com/app/graph_editor/

ε ε
i εi
j ε
εi j
j εi
εi i1
i1 ε
i i1
i1 j