An argument of A. Borel [Bor-61, Proposition 3.1] shows that every compact connected Lie group is homeomorphic to the Cartesian product of its derived subgroup and a torus. We prove a parallel result for definably compact definably connected groups definable in an o-minimal expansion of a real closed field. As opposed to the Lie case, however, we provide an example showing that the derived subgroup may not have a definable semidirect complement.
"Splitting definably compact groups in o-minimal structures." J. Symbolic Logic 76 (3) 973 - 986, September 2011. https://doi.org/10.2178/jsl/1309952529