LADW — The condition that ensures a real matrix $A$ to have real diagonalization

Question

In chapter 4 of Sergei Treil's Linear Algebra Done Wrong (LADW), the author first introduce the theorem that an operator $A: V \to V$ is diagonalizable iff for each eigenvalue $\lambda $, the geometric multiplicity coincides with its algebraic multiplicity. He then stated the following theorem as a corollary of the above theorem, which I dont see how they are correlated: $A$ real $n \times n$ matrix $A$ admits a real factorization (i.e., representation $A = S D S^{-1}$ where $S$ and $D$ are real matrices, $D$ is diagonal, and $S$ is invertible) if and only if it admits complex factorization and all eigenvalues of $A$ are real.