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Here is the octonion multiplication table drawn on a Möbius strip.
Each red triangle represents a quaternionic triplet.
A right-handed rotation around the triangle follows the multiplication identities.
So the triangle $e_1,e_2,e_3$ represents the identities $e_1\cdot e_2=e_3$, $e_2\cdot e_3=e_1$, and $e_3\cdot e_1=e_2$.

The vertices match up on either side of the Möbius strip.
On alternating sides are triangles skewed in opposing directions.