Definite forms and the Eichler-Kneser theorem

Definition A quadratic form is called indecomposable provided it cannot be expressed as the orthogonal sum of two non-trivial subspaces.

Theorem. A (positive) definite quadratic form has, up to ordering, a unique decomposition into indecomposables.

Proof. Given a positive definite quadratic form $Q=(L,C)$ and a decomposition $$Q=Q_1\perp \ldots \perp Q_l$$ into indecomposables, we have to prove uniqueness.
We call a nonzero $v\in L$ minimal, if it cannot be expressed as a sum $v=x+y$ with both $\langle x,x\rangle \lt \langle v,v\rangle$ and $\langle y,y\rangle \lt \langle v,v\rangle$. The procedure of expressing an element as a sum of two strictly shorter elements must stop after a finite number of steps. Hence the set $M\subset L$ of minimal elements generates $L$. On $M$, we define an equivalence relation by declaring $v \sim w$ if there exists a sequence $$v=x_1, x_2, \ldots, x_r=w$$ in $M$ with $\langle x_i,x_{i+1}\rangle\not= 0$ for $1 \leq i\lt r$.
A minimal element has to be contained in one of the summands $Q_k$ and any equivalent one has to be contained in the same summand. Together, equivalent minimal elements generate such a summand. So an indecomposable summand can be characterized as the span of an equivalence class of minimal elements.

Indecomposable definite quadratic forms have been classified up to rank 16. There are unique odd such in rank 1, 12, 14, 15 and 16 and unique even forms with rank 8 and rank 16. However, in higher ranks, the number of non-isomorphic such forms grows very rapidly. Considering only the relatively sparce even ones, there are exactly 22 of rank 24, more than 80 million of rank 32 (not classified yet) and more than $10^{51}$ of rank 40.