Commutation relations tell us which observables are compatible and which ones are not. How is that extended to more than two observables being measured at the same time (or successively)? If $\hat A$, $\hat B$, and $\hat C$ are hermitian operators corresponding to physical observables such that $[\hat A,\hat B]=0$ and $[\hat A,\hat C]=0$ (an optional restriction), is there any commutation relation that tells us about the compatibility of these observables? More briefly, how does the uncertainty principle extend beyond just 2 observables? Is it even possible to extend it to more than two observables? It doesn't seem unreasonable to want to measure more than two observables simultaneously.
You can check the commutations of every possible pair of observables in an arbitrarily long list, only if all pairs have $0$ commutator can you know the entire set of observables precisely.
For instance, in terms of angular momentum, you can know $L^2$ and one of: $L_x$, $L_y$ and $L_z$ precisely, in principal.
Is this what you mean?