The flow matrix is antisymmetric

flow_antisymm. For all real numbers $c, s$ (no hypotheses on them, so $c = s = 0$ is included), the $8\times 8$ real matrix $F(c,s)$ defined below is skew-symmetric:

The underlying bilinear product on $\mathbb R^8$. Index coordinates by $\{0,1,\dots,7\}$ with standard basis $e_0,\dots,e_7$. Relabel the indices $1,\dots,7$ as points of $\mathbb Z/7$ via $\pi(i) = (i+6) \bmod 7$, i.e. $\pi(i) = i-1$ for $1 \le i \le 7$ (also $\pi(0)=6$, but index $0$ never reaches the rule where $\pi$ is used). The "lines" are the seven 3-element sets $L_\ell = \{\ell,\ \ell+1,\ \ell+3\} \subseteq \mathbb Z/7$ for $\ell \in \mathbb Z/7$ (arithmetic mod $7$). An integer structure constant $T(i,j,k)$ (the $e_k$-coefficient of $e_i e_j$) is defined by the first applicable clause:

1. if $i = 0$: $T = 1$ if $j = k$, else $0$;
2. else if $j = 0$: $T = 1$ if $i = k$, else $0$;
3. else if $i = j$: $T = -1$ if $k = 0$, else $0$;
4. else if $k = 0$: $T = 0$;
5. else if there is $\ell$ with $L_\ell = \{\pi(i),\pi(j),\pi(k)\}$ and $(\pi(i),\pi(j))$ is one of $(\ell,\ell+1)$, $(\ell+1,\ell+3)$, $(\ell+3,\ell)$: $T = +1$;
6. else if there is $\ell$ with $L_\ell = \{\pi(i),\pi(j),\pi(k)\}$: $T = -1$;
7. otherwise $T = 0$.

(In clauses 5–6, if $k$ equals $i$ or $j$ the set $\{\pi(i),\pi(j),\pi(k)\}$ has at most two elements and cannot be a line, so $T=0$.) The product of $p, q \in \mathbb R^8$ is

Thus $e_0$ is a two-sided identity, $e_i e_i = -e_0$ for $i \ne 0$, and for distinct nonzero $i,j$, $e_i e_j = \pm e_k$ where $\pi(k)$ is the third point of the line through $\pi(i),\pi(j)$, with sign $+$ exactly when $(\pi(i),\pi(j))$ follows the cyclic order $\ell \to \ell+1 \to \ell+3 \to \ell$ on that line.

The matrix. For $a \in \mathbb R^8$, $R_a$ is the matrix with $(R_a)_{kj} = (e_j\cdot a)_k$ (the matrix of right multiplication $x \mapsto x\cdot a$), and $L_b$ has $(L_b)_{kj} = (b\cdot e_j)_k$ (the matrix of left multiplication $x \mapsto b\cdot x$). Then

The theorem asserts that, for every $c,s\in\mathbb R$, this matrix equals the negative of its transpose (equivalently all diagonal entries vanish and $F_{jk} = -F_{kj}$ for $j \ne k$). The Steiner-triple-system structure and other imported auxiliary definitions do not appear in the statement; only the line sets $L_\ell$ are used.