Find necessary and sufficient conditions on the quadratic form $ax^2+2bxy+cy^2$ on $\R^2$ such that the corresponding symmetric bi-linear form defines a Lorentz product on $\R^2$.
The matrix of the Gramian of the canonical basis $e_1,e_2$ of $\R^2$ is given by
$$
\left(
\begin{array}{cc}
a&b\\
b&c
\end{array}
\right)
$$
and the determinant is strictly negative iff $ac-b^2 < 0$.