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