Isometries between integral lattices (reviewed)

Two integral lattices, $L$ and $L'$, are isometric if there exists a map $\sigma:L\rightarrow L'$ such that $Q(\vec{v})=Q(\sigma(\vec{v}))$ for all $\vec{v}\in L$. If this is the case, we say $L\cong L'$. The set of all lattices $M$ such that $M\cong L$ is called the isometry class (or simply the class) of $L$.

If $L$ and $L'$ have associated Gram matrices $G$ and $G'$, then $L\cong L'$ implies that $G=T\cdot G'\cdot T^t$ for some $T\in GL_n(\mathbb Z)$.