I'll write $\mathbf{Z}^n$ for the integral quadratic form $x_1^2 + \cdots + x_n^2$. For which values of $n$ is $\mathbf{Z}^n$ unique in its genus, i.e. isolated in Kneser's graph? In particular can someone verify or shoot down the following guesses:

1. For $n \leq 8$, any lattice in the same genus as $\mathbf{Z}^n$ is isomorphic to $\mathbf{Z}^n$.

2. Any lattice in the same genus as $\mathbf{Z}^9$ is isomorphic to either $\mathbf{Z}^9$ or to $\mathbf{Z} \times \mathrm{E}_8$.