I have an operator $C$ that I wish to diagonalize on a Riemmanian manifold $M$ with constant curvature $\Lambda$ $$C = A + B$$ Now I know that these operators $A$ and $B$ commute in flat space, but on a curved space, they give $$[A,B] = \Lambda B$$
Does this mean that $C$ is not diagonalizable on $M$, since $A$ and $B$ aren't simultaneously diagonalizible?