When I read an answer in: An element in $\mathrm{SL}(2,\mathbb{R})$, I find a gap in the second proof (Olivier Bégassat's) of this question. Since my reputation points are not suffencient to comment, I asked here as a new question.
I think the second proof in this link may have some problem. Although two real matrices which that are conjugate as complex matrices are conjugate as real matrices, but the transformation matrices may not in $\mathrm{SL}(2,\mathbb{R})$. For example, $ \begin{pmatrix} 1&-1\\ 0&1\\ \end{pmatrix}$ is conjugate to $ \begin{pmatrix} 1&1\\ 0&1\\ \end{pmatrix}$ by $ \begin{pmatrix} -\mathrm{i}&0\\ 0&\mathrm{i}\\ \end{pmatrix}$, but it can not be conjugated in $\mathrm{SL}(2,\mathbb{R})$.
Is there someting wrong in his proof? I think conjugate in complex field and real field are equivalent can not imply the conclusion.
In fact, my own answer about this question is If any two $\mathrm{SL}(2,\mathbb{R})$ matrix can be conjugated by a matrix in $\mathrm{SL}(2,\mathbb{C})$, then they can be conjugated by a matrix in $\mathrm{PSL}(2,\mathbb{R})$.
Motivation: In many cases, we need the transformation of some objects preserve the structure, that is why I want the conjugate matrix is also in $\mathrm{SL}(2,\mathbb{R})$.
