Heads up: I'm a physicist.
I know that if $A=A^\dagger$, $B=B^\dagger$ commute ($AB=BA$) then there is an eigenbasis $\{v_1,v_2,...,v_n\}$ which diagonalises both $A$ and $B$ even if they are degenerate.
Is it always possible to find a common eigenbasis that is also orthonormal, i.e. $\langle v_i,v_j\rangle =\delta_{ij}$?
What if $A$ and $B$ are real symmetric?