Corollary: the frequencies are $0$, $2$ and $2\sin(\theta/2)$

Setting: the algebra on $\mathbb{R}^8$. Write $e_0,\dots,e_7$ for the standard basis of $\mathbb{R}^8$ (coordinates indexed by $\{0,\dots,7\}$). A bilinear product is defined by $e_i e_j = \sum_k T_{ijk}\, e_k$, i.e. for $p,q\in\mathbb{R}^8$, $(pq)_k = \sum_{i,j} p_i q_j T_{ijk}$, where the integer structure constants $T_{ijk}$ are given by the following rules, applied in this order:

• $e_0 e_j = e_j$ for all $j$, and $e_i e_0 = e_i$ for all $i$ (so $e_0$ is a two-sided unit, and $e_0e_0=e_0$);
• $e_i e_i = -e_0$ for $i = 1,\dots,7$;
• for distinct $i,j\in\{1,\dots,7\}$: the $e_0$-component of $e_ie_j$ is $0$; for $k\neq 0$, $T_{ijk}=\pm1$ if $\{i,j,k\}$ (as points of the Fano plane, see below) is a line, and $0$ otherwise. Hence $e_ie_j = \pm e_k$ with $k$ the third point of the unique line through $i$ and $j$.

The Fano plane here has points $\mathbb{Z}/7$, and $e_i$ ($1\le i\le 7$) is attached to the point $i-1 \bmod 7$ (via $i\mapsto (i+6)\bmod 7$). Lines are $\{l, l+1, l+3\}$ for $l\in\mathbb{Z}/7$. Translated back to basis indices $1,\dots,7$, the seven lines, each listed in its positive cyclic order, are

i.e. $(i,\,i+1,\,i+3)$ with indices taken in $\{1,\dots,7\}$ mod $7$. The sign is $+1$ exactly when $(i,j)$ is a consecutive pair in this cyclic order (that is, $(a,b)$, $(b,c)$ or $(c,a)$ for the line $(a,b,c)$), and $-1$ for the reversed order. So for each such triple $(a,b,c)$:

For example $e_1e_2=e_4$, $e_2e_1=-e_4$, $e_1e_3 = -e_7$, $e_7e_1=e_3$.

The matrices. For $a\in\mathbb{R}^8$, $R_a$ is the $8\times 8$ real matrix with $(k,j)$ entry $(e_j a)_k$: its $j$-th column is $e_j a$, so $R_a x = xa$ for every $x\in\mathbb{R}^8$ (right multiplication by $a$ in the standard basis). Likewise $L_b$ has $(k,j)$ entry $(b e_j)_k$, so $L_b x = bx$ (left multiplication by $b$). For reals $c,s$,

the matrix, in the standard basis (columns = images of basis vectors), of the linear map

The statement. For every real $\theta$ with $0<\theta<\pi$ (strict on both sides, so $\theta=0$ and $\theta=\pi$ are excluded), put $M = F(\cos\theta,\sin\theta)$, the matrix of $x\mapsto x e_1 + (\cos\theta\, e_1+\sin\theta\, e_2)\,x$. Let $\chi_M(X)=\det(X I - M)\in\mathbb{R}[X]$ be its characteristic polynomial, regarded in $\mathbb{C}[X]$ via $\mathbb{R}\subset\mathbb{C}$. The theorem asserts the conjunction of three claims:

1. The multiset of complex roots of $\chi_M$, counted with multiplicity (i.e. the eigenvalues of $M$ over $\mathbb{C}$ with algebraic multiplicities, 8 in total since $\chi_M$ is monic of degree 8), equals the multiset

(here $\sin\frac\theta2$ is the real number embedded in $\mathbb{C}$). Equivalently, $\chi_M(X) = X^2(X^2+4)\big(X^2+4\sin^2\tfrac\theta2\big)^2$ as a complex (hence real) polynomial; the multiplicities asserted are exactly: $0$ twice, $2i$ once, $-2i$ once, $2i\sin\frac\theta2$ twice, $-2i\sin\frac\theta2$ twice (with repeats merging if any of these values coincide). 2. $0 < \sin\frac{\theta}{2}$. 3. $\sin\frac{\theta}{2} < 1$.

There are no other hypotheses; $\theta$ is the only variable, and the parameters $c=\cos\theta$, $s=\sin\theta$ make $c\,e_1+s\,e_2$ satisfy $c^2+s^2=1$.