The seven lines

Which class is which depends on the marking. The outer automorphism exchanges each class $a$ with its class $b$, so the points and the lines of the Fano plane are assigned their classes only once the group of the plane is identified with the group of the projective line. The seam table and this journal’s labels use the identification of Seams, under which the points are class $a$. The volume’s own chart, its table of observers and pairs in Chapter X, uses the other: there the stabilizer of the clock 1 fixes the bisection $\{0,1,2,4\}\mid\{3,5,6,\infty\}$, which lies in class $b$. Read in that chart, the names exchange: the clocks are class $b$ and the lines class $a$, and so on for $A_4$ and $V_4$.

The lines of the Fano plane, and its complete quadrangles; the bisection $\{0,1,2,4\}|\{3,5,6,\infty\}$ of $\Proj^1(\F_7)$ and its orbit, and the perfect matchings in the orbit of $\{01,25,3\infty,46\}$; the subgroups $V_4^b$, $A_4^b$ and $S_4^b$; the conics for $\bar\alpha$ and the self-polar triangles from $V_4^b$ in Klein’s plane; the $K_4$’s of Coxeter vertices whose antiflags have a common line; and the quaternion subalgebras of the octonions.

In Thurston’s congruence link complement it is the complementary pairs of tetrahedra of class $b$, the lines of the Fano plane of the cells. In the lattice $E_8$ of class $a$, modulo $\bar\lambda$, the only invariant plane has nonzero vectors of class $S_4^b$, the module of the lines, as do the points of Klein’s $\mathbf 3$ reduced at the prime over $\bar\lambda$.