Semisimple algebraic groups over real closed fields

We give a self-contained introduction to linear algebraic and semialgebraic groups over real closed fields, and we generalize several key results about semisimple Lie groups to algebraic and semialgebraic groups over real closed fields. We prove that a torus in a semisimple algebraic group is maximal $\mathbb{R}$-split if and only if it is maximal $\mathbb{F}$-split for real closed fields $\mathbb{F}$. For the $\mathbb{F}$-points we formulate and prove the Iwasawa-decomposition $KAU$, the Cartan-decomposition $KAK$ and the Bruhat-decomposition $BWB$. For unipotent subgroups we prove the Baker-Campbell-Hausdorff formula, facilitating the analysis of root groups. We give a proof of the Jacobson-Morozov Lemma about subgroups whose Lie algebra is isomorphic to $\mathfrak{sl}_2$ for algebraic groups and a version for the $\mathbb{F}$-points, when the root system is reduced. We describe the rank 1 subgroups which are the semisimple parts of Levi-subgroups. We prove a semialgebraic version of Kostant's convexity theorem. The main tool used is a model theoretic transfer principle that follows from the Tarski-Seidenberg theorem.