Octonions
\(P_{0}: (a,b)(c,d)=(ac-d^*b,da+bc^*)\)	\(\uparrow \)	\(\circlearrowleft\)	\(\rightarrow\)
\(P_{1}: (a,b)(c,d)=(ac-db^*,a^*d+cb)\)	\(\downarrow \)	\(\circlearrowright\)	\(\leftarrow\)
\(P_{2}: (a,b)(c,d)=(ca-b^*d,da^*+bc)\)	\(\downarrow\)	\(\circlearrowleft\)	\(\rightarrow\)
\(P_{3}: (a,b)(c,d)=(ca-bd^*,ad+c^*b)\)	\(\uparrow\)	\(\circlearrowright\)	\(\leftarrow\)
Not Octonions (but similar)
\(Q_{0}: (a,b)(c,d)=(ca-d^*b,da+bc^*)\)	\(\uparrow\)	\(\circlearrowright\)	\(\rightarrow\)
\(Q_{1}: (a,b)(c,d)=(ca-db^*,a^*d+cb)\)	\(\downarrow \)	\(\circlearrowleft\)	\(\leftarrow\)
\(Q_{2}: (a,b)(c,d)=(ac-b^*d,da^*+bc)\)	\(\downarrow \)	\(\circlearrowright\)	\(\rightarrow\)
\(Q_{3}: (a,b)(c,d)=(ac-bd^*,ad+c^*b)\)	\(\uparrow \)	\(\circlearrowleft\)	\(\leftarrow\)
\(\downarrow\qquad\)Vertex to base
\(\uparrow\qquad\)Base to vertex
\(\circlearrowleft\qquad\)Traverse circle counter-clockwise
\(\circlearrowright\qquad\)Traverse circle clockwise
\(\rightarrow\qquad\)Traverse bases counter-clockwise
\(\leftarrow\qquad\)Traverse bases clockwise