Let's use the Schwarzschild spacetime notebook as an example, using these lines:

```
{r,t,\phi,\theta}::Coordinate;
{\mu,\nu,\rho,\sigma,\lambda,\kappa,\chi,\gamma}::Indices(values={t,r,\phi,\theta}, position=fixed);
\partial{#}::PartialDerivative;
g_{\mu\nu}::Metric.
g^{\mu\nu}::InverseMetric.
ss:= { g_{t t} = -(1-2 M/r), g_{r r} = 1/(1-2 M/r), g_{\theta\theta} = r**2, g_{\phi\phi}=r**2 sin(\theta)**2 }.
complete(ss, $g^{\mu\nu}$);
```

So, we have $g_{\mu \nu}$ and $g^{\mu \nu}$ completely determined and we should be able to explicitly calculate its determinant and trace. How would one do it properly in Cadabra?