The octonion product from the Fano plane, and the flow matrix $R_{e_1} + L_{c e_1 + s e_2}$

Conventions and imported definitions. Indices range over $\{0,1,\dots,7\}$ (the type of 8 elements); write $e_0,\dots,e_7$ for the standard basis of $\mathbb{R}^8$ (the vector with a $1$ in coordinate $j$ and $0$ elsewhere is $e_j$). The imported Fano plane uses points $\mathbb{Z}/7 = \{0,\dots,6\}$ with arithmetic mod 7, and for $l \in \mathbb{Z}/7$ the line $L_l = \{l,\ l+1,\ l+3\}$ (a finite set). These seven sets are pairwise distinct, each has 3 elements, and every two distinct points lie on exactly one of them. The structure fano built from them is not used by the definitions below; only $L_l$ is.

fanoPoint. For $i \in \{0,\dots,7\}$,

$$\pi(i) = (i+6) \bmod 7 \in \mathbb{Z}/7 .$$

So $\pi(m) = m-1$ for $m = 1,\dots,7$, and also $\pi(0) = 6 = \pi(7)$. This map is not injective, but $\pi(0)$ is never consulted by octTable (see below), because every case involving index $0$ is settled before the Fano test.

octTable. An integer $T(i,j,k)$ for $i,j,k \in \{0,\dots,7\}$, defined by the first applicable case:

1. if $i = 0$: $T = 1$ if $j = k$, and $0$ otherwise;
2. else if $j = 0$: $T = 1$ if $i = k$, and $0$ otherwise;
3. else if $i = j$ (both nonzero): $T = -1$ if $k = 0$, and $0$ otherwise;
4. else if $k = 0$: $T = 0$;
5. else ($i \ne j$, all of $i,j,k$ nonzero): $T = +1$ if there is an $l \in \mathbb{Z}/7$ with $L_l = \{\pi(i),\pi(j),\pi(k)\}$ as sets and $(\pi(i),\pi(j))$ equal to one of $(l,l+1)$, $(l+1,l+3)$ or $(l+3,l)$. Otherwise $T = -1$ if some $L_l$ equals $\{\pi(i),\pi(j),\pi(k)\}$. Otherwise $T = 0$.

In case 5, the set $\{\pi(i),\pi(j),\pi(k)\}$ can only equal a line if it has three elements, which means $k \notin \{i,j\}$. The ordered pair $(\pi(i),\pi(j))$ gets $+1$ when it follows the cyclic order $l \to l+1 \to l+3 \to l$ of its line and $-1$ for the reverse order. With $T(i,j,k)$ read as the coefficient of $e_k$ in $e_i e_j$, the table says that $e_0$ is a two-sided identity, that $e_m^2 = -e_0$ for $m = 1,\dots,7$, and that for distinct $i,j \in \{1,\dots,7\}$, $e_ie_j = \pm e_k$ with $k$ the unique third index on their line. Using $\pi(m) = m-1$, the lines correspond to the index triples $\{m, m+1, m+3\}$ (indices taken mod 7 in $\{1,\dots,7\}$), and the $+1$ products are exactly

$$\begin{aligned} &e_1e_2=e_4,\ e_2e_4=e_1,\ e_4e_1=e_2; &&e_2e_3=e_5,\ e_3e_5=e_2,\ e_5e_2=e_3;\\ &e_3e_4=e_6,\ e_4e_6=e_3,\ e_6e_3=e_4; &&e_4e_5=e_7,\ e_5e_7=e_4,\ e_7e_4=e_5;\\ &e_5e_6=e_1,\ e_6e_1=e_5,\ e_1e_5=e_6; &&e_6e_7=e_2,\ e_7e_2=e_6,\ e_2e_6=e_7;\\ &e_7e_1=e_3,\ e_1e_3=e_7,\ e_3e_7=e_1, \end{aligned}$$

that is, $e_me_{m+1} = e_{m+3}$ and its cyclic shifts. Reversing the order of the factors in any of these gives the negative (for example $e_2e_1 = -e_4$). All other coefficients $T(i,j,k)$ are $0$.

omul. For $p, q \in \mathbb{R}^8$ (arbitrary real 8-tuples, with no normalisation), $p \cdot q \in \mathbb{R}^8$ is defined coordinatewise by

$$(p\cdot q)_k = \sum_{i=0}^{7}\sum_{j=0}^{7} p_i\, q_j\, T(i,j,k),$$

with $T$ cast from $\mathbb{Z}$ to $\mathbb{R}$. This is the $\mathbb{R}$-bilinear product on $\mathbb{R}^8$ whose values on basis vectors are $e_i \cdot e_j = \sum_k T(i,j,k)\, e_k$, as listed above.

Rmat. For $a \in \mathbb{R}^8$, $R_a$ is the real $8\times 8$ matrix with rows and columns indexed by $\{0,\dots,7\}$ and entries

$$(R_a)_{k j} = (e_j \cdot a)_k \qquad (k = \text{row},\ j = \text{column}).$$

Column $j$ is the coordinate vector of $e_j\cdot a$, so for column vectors $x$ we have $R_a x = x\cdot a$. In other words, $R_a$ is the matrix of right multiplication by $a$ in the standard basis.

Lmat. For $b \in \mathbb{R}^8$, $L_b$ is the $8\times 8$ matrix with $(L_b)_{kj} = (b\cdot e_j)_k$ (row $k$, column $j$). Column $j$ is $b\cdot e_j$, so $L_b x = b \cdot x$. In other words, $L_b$ is the matrix of left multiplication by $b$ in the standard basis.

flowMat. For arbitrary reals $c, s$ (with no constraint such as $c^2+s^2=1$; they may be any real numbers, including $0$),

$$F(c,s) = R_{e_1} + L_{c e_1 + s e_2},$$

where $c e_1 + s e_2$ is the vector with coordinate $1$ equal to $c$, coordinate $2$ equal to $s$, and all others $0$. So $F(c,s)$ is the matrix, acting on column vectors in the standard basis, of the linear map

$$x \mapsto x\cdot e_1 + (c e_1 + s e_2)\cdot x .$$

Here $e_1, e_2$ are the basis vectors in positions 1 and 2, the first two imaginary units, not $e_0$. Its columns (the images of $e_0,\dots,e_7$), obtained by expanding with the table above, are

$$\begin{aligned} e_0 &\mapsto (1+c)\,e_1 + s\,e_2, & e_1 &\mapsto -(1+c)\,e_0 - s\,e_4,\\ e_2 &\mapsto -s\,e_0 + (c-1)\,e_4, & e_3 &\mapsto s\,e_5 + (c-1)\,e_7,\\ e_4 &\mapsto s\,e_1 + (1-c)\,e_2, & e_5 &\mapsto -s\,e_3 + (c-1)\,e_6,\\ e_6 &\mapsto (1-c)\,e_5 + s\,e_7, & e_7 &\mapsto (1-c)\,e_3 - s\,e_6. \end{aligned}$$