Orthogonal decomposition of a lattice (awaiting review)

Let $L$ be an integral lattice, with bilinear form $B : L \times L \to \Z$. An orthogonal decomposition of $L$ is a direct sum decomposition

$$ L = L_1 \oplus L_2 \oplus \dots \oplus L_r $$

such that $B(x_i, x_j) = 0$ for all $x_i \in L_i$, $x_j \in L_j$, where $i \not = j$. Equivalently, if $G$ is the Gram matrix of $L$, then after a suitable change of basis, it has the block diagonal form

$$ G = \begin{pmatrix} G_1 & & & \\ & G_2 & & \\ & & \ddots & \\ & & & G_r \end{pmatrix} $$

where each $G_i$ is a Gram matrix for $L_i$.

A lattice is (orthogonally) indecomposable if it does not admit a non-trivial orthogonal decomposition.

A theorem of Kneser states that every lattice admits a unique orthogonal decomposition into indecomposable summands. This decomposition is unique up to ordering and isometry of the summands.