The object of size 14, class a

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 oriented quadrilaterals of the Fano plane; the four-subsets of $\Proj^1(\F_7)$ in the orbit of $\{0,1,2,5\}$; the groups $V_4^a$ with a cyclic order of their involutions; the oriented self-polar triangles from $V_4^a$; the oriented $K_4$’s of Coxeter vertices whose antiflags have a common point; and the 14 ideal tetrahedra of Thurston’s congruence link complement whose cusp sets lie in the orbit of $\{0,1,2,5\}$.

The 28 ideal tetrahedra form one orbit of $\PGL(2,7)$, with stabilizer $A_4$, and two orbits of 14 under $\PSL(2,7)$, the objects of classes $a$ and $b$; the elements of $\PGL(2,7)$ outside $\PSL(2,7)$ exchange them. Read through the Fano plane, the tetrahedra are the complete quadrangles and quadrilaterals, each with one of its two orientations.

In the lattice $E_8$ that $\SL(2,7)$ preserves for the tetrahedra of class $a$, a lattice over $\Z[\lambda]$, $\lambda=(1+\sqrt{-7})/2$, the sixteen vectors lie in one residue modulo $\lambda$, and the fourteen other nonzero residues form one orbit with stabilizers of class $A_4^a$. Modulo $\bar\lambda$ the sixteen vectors are the eight points of an affine 3-space whose planes are the tetrahedra of class $a$. The class $A_4^a$ does not occur among the thirty octonion lattices.

That lattice $E_8$, over the integers of $K=\Q(\sqrt{-7})$, is classical: its unitary group $2{\cdot}A_7$ acts on $K^4$ by a faithful irreducible character, the invariant lattice has an even unimodular trace form, so it is $E_8$ by uniqueness, and the group of order 168 is the stabilizer of one cross.