Stability of the E8-form (Ex)

Show there exists an isometry of unimodular forms

$\langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle -1 \rangle \cong ({\mathbb Z}^8,\lambda) \oplus \langle -1 \rangle$$<wikitex>; Show there exists an isometry of unimodular forms $\langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle -1 \rangle \cong ({\mathbb Z}^8,\lambda) \oplus \langle -1 \rangle$ where $\lambda$ is unimodular, even, and positive definite. (There is a unique such form in dimension 8: extra credit is given for an isometry $({\mathbb Z}^8,\lambda) \cong E_8$). </wikitex> [[Category:Exercises]] [[Category:Exercises without solution]]\langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle 1 \rangle \oplus \langle -1 \rangle \cong ({\mathbb Z}^8,\lambda) \oplus \langle -1 \rangle$

where $\lambda$$\lambda$ is unimodular, even, and positive definite. (There is a unique such form in dimension 8: extra credit is given for an isometry $({\mathbb Z}^8,\lambda) \cong E_8$$({\mathbb Z}^8,\lambda) \cong E_8$).