Both statements are well-known in the arithmetic theory of quadratic forms (aka integral lattices).

The first statement is a consequence of Hermite's bound on the nonzero minimum of an integral lattice: the minimum of a unimodular lattice of rank $\leq 8$ is 1.

The second statement is a consequence of Kneser's theory of neighbors. You can find it in the last theorem in O'Meara's book. It also mentioned that Kneser completed the analysis of the genus of $I_n$ for $n \leq 13$.

Both statements are well-known in the arithmetic theory of quadratic forms (aka integral lattices).

The first statement for $n \leq 5$ is a consequence of Hermite's bound on the nonzero minimum of an integral lattice: the minimum of a unimodular lattice of rank $\leq 5$ is 1.

Both statements are consequences of Kneser's theory of neighbors. You can find it in the last theorem in O'Meara's book. It also mentioned that Kneser completed the analysis of the genus of $I_n$ for $n \leq 13$.