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My main confusion is this one:

##L^2## commutes with ##L_i##. Then these two share eigenstates. But ##L_x## and ##L_y## do not commute, so they don't share eigenstates. Isn't this violating some type of transitivity? Namely, if ##[L^2,L_i]=0## then exactly what accounts for ##L_x## and ##L_y## not sharing eigenstates (considering they share eigenstates with ##L^2##)?

A mathematical explanation of this would be much appreciated.

If my confusion is not well understood please ask for clarification.

Many thanks.