If $S$ and $T$ are commuting, normal operators, then $ST$ is normal
That says it all, but let me be more specific. (By the way Wikipedia says this: "The product of normal operators that commute is again normal; this is nontrivial and follows from Fuglede's theorem")
Let $V$ be a finite dimensional inner-product space. Do not assume that it is complex. Suppose that $S$ and $T$ are normal operators that commute. Prove that $ST$ is normal.
This result is relatively simple if we had assumed that our inner-product space was complex. Just by using the spectral theorem, we can simultaneously diagonalize $S$ and $T$ and we would be done.
The question is, then, how does one show this for a real inner-product space. One suggestion is to complexify our space --- but I am not too familiar with this. Does anyone care to give this a shot?