Using the Fano plane for octonion multiplication The Fano plane is the projective plane over the field $\mathbf Z/2$.
It can be used to remember octonion multiplication, as nicely explianed in John Baez's article on octonions (see http://math.ucr.edu/home/baez/octonions/).
The picture (taken from Baez's website) is as follows: 

It indicates for example, using the cyclic orderings on the lines, that $e_6 \cdot e_1= e_5$ but that $e_6 \cdot e_4= -e_3$.  
Two natural questions arise for me: 


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*Why did Baez label the circles in such a weird order. There is probably a good reason which is implicit. (A priori, if I permute the labels arbitrarily, I'll get something isomorphic).

*Also, one could decide to choose other orientations for the arrows.
So again, why did Baez choose these orientations? And what could we get with other orientations? 
 A: The circles are labeled in such a way that the lines are given by $(i,i+1,i+3)$ modulo $7$. (Interpreted in the interval $1,...,7$). This is with good reason, see below.
The orientations are more difficult to explain in a sentence, but two remarks are in order:


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*They are drawn in such a way that the automorphisms group $G\cong \mathsf{PSL}(3,2)\cong\mathsf{PSL}(2,7)$ of the Fano plane will induce automorphisms of the octonions. For instance, invariance under rotations (of oder $3$) is immediately visible from the picture; an element of order $7$ in $G$ is given by mapping $e_i\to e_{i+1}$ and since the orientation is always $(i,i+1,i+3)$ the arrows are invariant under these automorphisms too. (The automorphisms of order $2$ correspond to reflections the Fano plane, but you also have to introduce some signs like $e_i\mapsto -e_j$ so this is less obvious.) This is somewhat useful since it will immediately provide you with a bunch of automorphisms of the octonions.

*You could in principle draw arbitrary arrows and study the resulting algebra generated by the same mechanism. But it is a fact that if you want the outcome to be a composition algebra, then the quadratic form determines the octonion algebra entirely. (This quadratic form is a so called Pfister form, and I think this theorem is usually attributed to Pfister. [The theorem is actually due to Jacobson, see the comment by Mariano Suárez-Alvarez here. ]) So in some sense, if you specify what $e_i^2$ is (usually one studies $e_i^2=-1$ for all $i$, this corresponds to the so called "compact real octonions") then you have specified the quadratic form entirely and there is essentially only one way to orient the edges to obtain a composition algebra.

*It's very well possible that drawing different orientations on the edges gives rise to other algebra's with interesting properties.  I would be very interested if anyone knew of results in that direction.

