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

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

Since \(a = r\), \(L^* = \tfrac{1}{2} L\), and \(L\) is the twist \(M(2)\) of the unimodular lattice \(M = U \oplus E_{8}(-1)\), the record 0128. The corpus records a twist \(M(n)\), \(|n| \geq 2\), only for the 13 rows \(a = r\) of Nikulin’s table, because the classification names the twist and not \(M\).

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.

\(U\) is the lattice with basis \(e, f\) and \(b(e, e) = b(f, f) = 0\), \(b(e, f) = 1\). \(L(k)\) is the module \(L\) with the form \(k b\). \(E_8\) is the root lattice of that type: \(e_1, \dots, e_8\) is a basis of simple roots, numbered as SageMath’s CartanMatrix numbers them, and \(b(e_i, e_j)\) is the entry \((i, j)\) of the Cartan matrix. The basis of an orthogonal sum is the union of the bases of the summands, in the order written.