The pseudo-metric, such as those used in general relativity, can be expressed as the following sum:
$$ ds^2=\sum_{\nu}\sum_{mu}g_{\nu\mu}dX_\nu dX_{\mu} $$
The elements $g_{\nu\mu}$ can be organized as a matrix
$$ g=\pmatrix{g_{00} & g_{01} & g_{02} & g_{03} \\ g_{10} & g_{11} & g_{12} & g_{13} \\ g_{20} & g_{21} & g_{22} & g_{23} \\ g_{30} & g_{31} & g_{32} & g_{33} } $$
The only restriction I am aware of is that all elements of $g$ must be functions of two vectors of the tangent space of the manifold to the real numbers : $g_{\nu\mu}: V \times V \to \mathbb{R}$. Otherwise, the value $\mathbb{R}$ can be $0$, negative or positive. This does not seem true to me for the following reasons:
Consider that $ds^2$ is a pseudo-metric, formally defined for all $x,y,z$ elements of the manifold $M$,
$$ s(x,x)=0\\ s(x,y)=s(y,x)\\ s(x,y)\leq s(x,y) + s(y,z) $$
The last requirement, the triangle inequality seems to me, intuitively, to imply additional restrictions. It seems to me that the cross-elements cannot exceed some value. Explicitely, from the triangle inequality, does it follow that the coefficient of cross terms cannot exceed inequalities of this type:
$$ g_{01}+g_{10} \leq 2g_{00}g_{11} \\ g_{02}+g_{20} \leq 2g_{00}g_{22} \\ g_{03}+g_{30} \leq 2g_{00}g_{33} \\ g_{21}+g_{12} \leq 2g_{11}g_{22}\\ \vdots $$
If these inequalities are violated, the triangle inequality is also violated.