In the basis $(e_0, e_1, e_2, e_4 \mid e_3, e_5, e_6, e_7)$ the flow matrix is block diagonal

Statement. For all real numbers $c, s \in \mathbb{R}$ (no other hypotheses: in particular no relation such as $c^2+s^2=1$ is assumed, and $c,s$ may be $0$), the $8\times 8$ real matrix $F_{c,s}$ defined below, after its rows and columns are simultaneously permuted by the index bijection $\beta$ below, equals the block-diagonal matrix

with the explicit $4\times 4$ blocks $A_{c,s}$, $B_{c,s}$ given below and $4\times 4$ zero off-diagonal blocks.

The algebra. Let $e_0,\dots,e_7$ be the standard basis of $\mathbb{R}^8$ (indices $0,\dots,7$). A bilinear product on $\mathbb{R}^8$ is defined by $p\cdot q = \sum_{i,j,k} p_i\, q_j\, T(i,j,k)\, e_k$, where the integer structure constants $T(i,j,k)$ (so $e_i e_j = \sum_k T(i,j,k) e_k$) are:

• $e_0 e_j = e_j$ for all $j$; $e_i e_0 = e_i$ for all $i$ (the first rule wins when $i=0$, giving $e_0e_0=e_0$);
• $e_i e_i = -e_0$ for $i \in \{1,\dots,7\}$;
• for distinct $i,j \in\{1,\dots,7\}$: the $e_0$-coefficient is $0$; and for $k\in\{1,\dots,7\}$, relabel $i\mapsto \pi(i) = (i+6) \bmod 7 = i-1 \in \mathbb{Z}/7$. The Fano lines are the sets $\{l,\,l+1,\,l+3\}\subset\mathbb{Z}/7$, $l\in\mathbb{Z}/7$. Then $T(i,j,k)=+1$ if $\{\pi(i),\pi(j),\pi(k)\}$ equals some line $\{l,l+1,l+3\}$ and $(\pi(i),\pi(j))$ is one of $(l,l+1)$, $(l+1,l+3)$, $(l+3,l)$; $T(i,j,k)=-1$ if $\{\pi(i),\pi(j),\pi(k)\}$ is a line but the ordering is not of that form; $T(i,j,k)=0$ otherwise (in particular when $k\in\{i,j\}$, since then the set has only 2 elements).

Concretely, in terms of basis indices, $e_ie_j=e_k$ for $(i,j,k)$ any cyclic rotation of

and $e_je_i=-e_k$ for these, all other products of distinct imaginary units having zero coefficient.

The matrix $F_{c,s}$. For $a,b\in\mathbb{R}^8$, $R_a$ is the matrix with entry $(k,j)$ equal to the $k$-th coordinate of $e_j\cdot a$, i.e. the matrix (acting on column vectors, column $j$ = image of $e_j$) of right multiplication $x\mapsto x\cdot a$ in the standard basis; $L_b$ is likewise the matrix of left multiplication $x\mapsto b\cdot x$. Then

the matrix of the linear map $x \mapsto x\,e_1 + (c\,e_1+s\,e_2)\,x$, with $(F_{c,s})_{k,j}$ = the $e_k$-coefficient of the image of $e_j$.

The bijection and reindexing. $\beta:\{0,\dots,7\}\to\{0,1,2,3\}\sqcup\{0,1,2,3\}$ sends $0,1,2,4$ to the first copy's $0,1,2,3$ and $3,5,6,7$ to the second copy's $0,1,2,3$ (so $\beta^{-1}$ of first-copy $i$ is $p_i$ with $(p_0,p_1,p_2,p_3)=(0,1,2,4)$, and of second-copy $i$ is $q_i$ with $(q_0,q_1,q_2,q_3)=(3,5,6,7)$). The reindexed matrix has entry $F_{c,s}(\beta^{-1}(x),\beta^{-1}(y))$ at position $(x,y)$, the same bijection being used for rows and columns. Hence the statement is equivalent to: for all $i,j\in\{0,1,2,3\}$,

i.e. the coordinate subspaces $\operatorname{span}(e_0,e_1,e_2,e_4)$ and $\operatorname{span}(e_3,e_5,e_6,e_7)$ are each mapped into themselves, with the stated matrices in those ordered bases.

The blocks.

The equation is an equality of real $8\times 8$ matrices indexed by $\{0,1,2,3\}\sqcup\{0,1,2,3\}$, required to hold identically in $c$ and $s$.