Given the gram matrix
$ \begin{bmatrix}
1&0\\
0&-1
\end{bmatrix}$
which is nothing but the one dimensional minkowski metrik.
the question states, find bases in $\mathbb{R^2} \times \mathbb{R^2}$
And considering the "unknown" bilinearform $b$ which belongs to this matrix. The question states, find basis $(v_1,v_2): \forall \: i \; b(v_i,v_i) > 0 ,b(v_i,v_i)<0, b(v_i,v_i)=0 $
Thus we conclude, that the definitness of a bilinear can not be examined n a basis.
so this is what i meant last night, especially, the last comment.