I was trying to find components of the Riemann tensor and it occurred to me that there could be an issue with my notation. For example, if one particular component of the tensor is
$$ R^{\theta}{}_{t\theta t}, $$
and I wanted to find the component $R^t{}_{\theta\theta t}$, then I could
$$ R^t{}_{\theta\theta t}=g^{tt}(-g_{\theta\theta} R^{\theta}{}_{t\theta t}); $$ i.e. lower the $\theta$ index with the $g_{\theta\theta}$ component of the metric, then use the fact that $R_{t\theta\theta t}=-R_{\theta t\theta t}$ and finally, raise the $t$ index with the $g^{tt}$ component of the inverse metric.
I know that there should only be two indices the same on any one term, but, in this case these indices are neither free nor being summed over. So, is this notation allowed and is it correct to use the metric in this way?