This is elementary, but to take a slightly different line from what I said in the first comment, the theory of the rational canonical form allows us to diagonalize a real square matrix with multiplicity free minimum polynomial by conjugating by an invertible real matrix $T.$ The only irreducible monic polynomials in $\mathbb{R}[x]$ are linear, or quadratic with a pair of complex conjugate roots. Each irreducible factor of degree $d$ of the characteristic polynomial of the matrix gives rise to $d \times d$ block. If $d = 1,$ the block is obvious. If $d = 2,$ and the monic factor is $x^{2} -2rx \cos \theta + r^{2}$ for positive real $r$ and $0 < \theta < 2 \pi,$ then the corresponding $2 \times 2$ block is $\left( \begin{array}{clcr} 0 & 1\\ -r^{2} & 2r\cos \theta \end{array} \right)$ which is a real matrix with eigenvalues $re^{ i \theta}$ and $re^{-i \theta}$ and is the companion matrix to the polynomial $x^{2} - 2rx \cos \theta + r^{2}.$

This is elementary, but to take a slightly different line from what I said in the comments, the theory of the rational canonical form allows us to diagonalize a real square matrix with multiplicity free minimum polynomial by conjugating by an invertible real matrix $T.$ The only irreducible monic polynomials in $\mathbb{R}[x]$ are linear, or quadratic with a pair of complex conjugate roots. Each irreducible factor of degree $d$ of the characteristic polynomial of the matrix gives rise to $d \times d$ block. If $d = 1,$ the block is obvious. If $d = 2,$ and the monic factor is $x^{2} -2rx \cos \theta + r^{2}$ for positive real $r$ and $0 < \theta < 2 \pi,$ then the corresponding $2 \times 2$ block is $\left( \begin{array}{clcr} 0 & 1\\ -r^{2} & 2r\cos \theta \end{array} \right)$ which is a real matrix with eigenvalues $re^{ i \theta}$ and $re^{-i \theta}$ and is the companion matrix to the polynomial $x^{2} - 2rx \cos \theta + r^{2}.$