\(U \oplus E_{8}(-2)\)

This is the even hyperbolic 2-elementary lattice with invariants \((r, a, \delta) = (10, 8, 0)\).

Nikulin lists it in Table 1 of Factor groups of groups of automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections (1983), the table of the even hyperbolic lattices \(L\) with \(2 A_L = 0\) that have a primitive embedding in the K3 lattice. These are the invariant lattices of the nonsymplectic involutions of K3 surfaces (Remark 4.5.3), and Theorem 4.3.2 states that \((r, a, \delta)\) determines each of them.

Roots: let \(e, f\) be the basis of \(U\), and let \(r = a e + c f + v\) be a root, with \(v\) in \(E_8(-2)\). Write \(v = m v'\) with \(v'\) primitive, and \(m = 0\) when \(v = 0\). \(E_8\) is unimodular, so \(b(v, E_8(-2)) = 2m\mathbb{Z}\), and \(b(r, L) = d\mathbb{Z}\) with \(d = \gcd(a, c, 2m)\). \(r\) is primitive, so \(\gcd(a, c, m) = 1\), and \(d \in \{1, 2\}\). \(b(r, r) = 2ac + b(v, v)\), and 4 divides \(b(v, v)\). \(b(r, r)\) divides \(2 b(r, x)\) for each \(x\) in \(L\), so it divides \(2d\). When \(d = 2\), \(a\) and \(c\) are even. When \(d = 1\), \(b(r, r)\) divides 2 and is even, so \(ac + b(v, v)/2 = \pm 1\), and \(a\) and \(c\) are odd. Each root is therefore in \(M = \{a \equiv c \bmod 2\}\), which has index 2 in \(L\) and the basis \(e + f\), \(e - f\), \(e_1, \dots, e_8\). Each vector of that basis is a root: \(\mathbb{Z}\Phi(L) = M = \mathbb{Z}(e + f) \oplus \mathbb{Z}(e - f) \oplus E_8(-2)\), an orthogonal sum.