Definition. Let $G \subset GL(n,\mathbb{R})$ be a group. $G$ is called $\mathbb{R}$-almost algebraic if there is an algebraic group $H$ defined over $\mathbb{R}$ such that the $\mathbb{R}$-points $H(\mathbb{R})$ of $H$ is a cocompact normal subgroup of $G$.

It is claimed in [Zimmer, Ergodic Theory and Semisimple Groups, Page 40] that any $\mathbb{R}$-almost algebraic group is the real points of an algebraic group defined over $\mathbb{R}$. I am trying to prove this claim but without any progress. Can anyone explain to me why this claim is true?

Thank you.