Let $P$ be minimal parabolic real Lie group of some non-compact semisimple Lie group $G$, then $P=MAN$ by Langlands decomposition where $A$ is abelian and $\mathbb{R}-$diagonalizable. Suppose $\rho:A\to \text{GL}(2,\mathbb{R})$ is a rational representation, is it true that the elements in the image will also be diagonalizable?
If the representation is not assumed to be rational, then I think the mapping $\text{PSL}(2,\mathbb{R})\to \text{GL}(2,\mathbb{R})$ $$\begin{pmatrix} e^\lambda & 0\\ 0 & e^{-\lambda}\end{pmatrix}\mapsto\begin{pmatrix} 1 & \lambda \\0 & 1 \end{pmatrix}$$ is an easy counterexample, and I had the impression that Shur's lemma shouldn't hold for real abelian groups. However I'm not confident in how much a role rational mappings will play.
The statement is correct per Proposition 8.2 of Borel's Linear Algebraic groups. Any rational representations of the $\mathbb{R}-$split tori is diagonalizable.