In proving the spectral theorem for self-adjoint operators, the first step is to show that an eigenvalue exists (and then you do induction).
Over $\mathbb{C}$, this is easy, since it's an algebraicly closed field.
Over $\mathbb{R}$, the books I've seen use a sort've long proof. But if you have a self-adjoint real matrix, then it is also a self-adjoint complex matrix. Therefore you can find eigenvalues, and you know those eigenvalues will be real because it's self-adjoint. Done. What am I overlooking?