  We develop the theory of algebraic groups over real closed fields and apply
the results to construct a geometric object $\mathcal{B}$ and to prove that
$\mathcal{B}$ is an affine $\Lambda$-building. We use a model theoretic
transfer principle to prove generalizations of statements about semisimple Lie
groups. In this direction we give proofs for the Iwasawa-decomposition $KAU$,
the Cartan-decomposition $KAK$ and the Bruhat-decomposition $BWB$. For
unipotent subgroups we prove the Baker-Campbell-Hausdorff formula and use it to
analyse root groups. We give a proof of the Jacobson-Morozov Lemma about
subgroups whose Lie algebra is isomorphic to $\mathfrak{sl}_2$ and we describe
other rank 1 subgroups which are the semisimple parts of Levi-subgroups. We
prove a semialgebraic version of Kostant's convexity.
  Over the reals, semisimple Lie groups are closely related to the symmetry
groups of symmetric spaces of non-compact type. These symmetric spaces can be
described semialgebraically, which allows us to consider their semialgebraic
extension over any real closed field. Starting from these non-standard
symmetric spaces we use a valuation (with image some non-discrete ordered
abelian group $\Lambda$) on the fields to define a $\Lambda$-pseudometric.
Identifying points of distance zero results in a $\Lambda$-metric space
$\mathcal{B}$. Assuming that the root system of the associated Lie group is
reduced, we prove that $\mathcal{B}$ is an affine $\Lambda$-building. The proof
relies on a thorough analysis of stabilizers.
