This is proved in the book *Representations of compact Lie groups* by Bröcker and tom Dieck and reviewed [here][1].  It is Proposition 7.5 in Chapter V.  The proof is for compact, connected Lie groups, but any connected Lie group has the homotopy type of its maximal compact subgroup.  (Everything here is for finite-dimensional Lie groups, of course.)



  [1]: http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183553688