Suppose $F$ is a free abelian group with finite rank $>1$ a symmetric bilinear form $B:F\times F \to \Bbb Z$. We say that $F$ is decomposable if there are nontrivial free abelian groups $F_1, F_2$ and bilinear forms $B_i$ on $F_i$ so that there is a group isomorphism $f:F_1\oplus F_2\to F$ such that $B(f(x_1,x_2),f(y_1,y_2))=B_1(x_1,y_1)+B_2(x_2,y_2).$ Suppose we are explicitly given $(F,B)$. Is there a general way to check whether it is decomposable or not?
Actually I’m on the following situation. Consider the matrix $$ A= \left( \begin{array}{rrrrrrrrrrrrrrr} 1 & - 1 & - 1 & - 1 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ - 1 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ - 1 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ - 1 & 0 & 0 & 7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ - 1 & 0 & 0 & 0 & 4 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 2 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 2 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 2 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 2 & - 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 2 & - 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 2 & - 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 2 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 2 & - 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 2 \\ \end{array} \right) $$ This is a positive definite symmetric matrix of determinant 1, and it defines a symmetric bilinear form $B$ on $\Bbb Z^{15}$ by $B(x,y)=x^t Ay$. I want to check whether $(\Bbb Z^{15},F)$ is decomposable or not, but I have no idea where to start.