Let $V$ be a finite-dimensional inner product space over $\mathbb{R}$.
Let $\mathscr{A}$ be a family of self-adjoint operators on $V$ such that $ST=TS$ for all $S,T\in \mathscr{A}$.
Then, does there exists an orthonormal basis $\beta$ for $V$ such that $[T]_{\beta}$ is diagonal for every $T\in\mathscr{A}$?
I'm going to prove this by induction: Decopmose $V$ into the direct sum of eigenspaces, then find an orthonormal basis for each eigenspace and take a union.
Before I really write down to prove this, I want to clarify whether this is true or not. (I don't want to waste time)..
Is this $\mathscr{A}$ simultaneously diagonalizable by an orthonormal basis?